Divine coincidence: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
No edit summary
 
 
Line 1: Line 1:
The '''trinomial triangle''' is a variation of [[Pascal's triangle]].  The difference between the two is that an entry in the trinomial triangle is the sum of the ''three'' (rather than the ''two'' in Pascal's triangle) entries above it:


<center>
<math>\begin{matrix}
& &  &  & 1\\
& &  & 1& 1&1\\
& & 1& 2& 3&2&1\\
&1& 3& 6& 7&6&3&1\\
1&4&10&16&19&16&10&4&1\end{matrix}</math>
</center>


The <math>k</math>-th entry of the <math>n</math>-th row is denoted by
Visiting at least three quotes sites raise the chances that you would realize more savings. On the one hand, you have people who are strong and mighty. But guaranteed issue policies  usually refund all premiums paid with interest even if the waiting period has not been satisfied. Look for the carriers that are rated A+ or better by the A. A lot of agencies out there are going to make some determined sales phone calls to arrange an appointment with you. In the unthinkable event of your passing, the right life insurance coverage will be absolutely crucial to your family's future well-being. You can do it all online, and there is no need of an agent.  <br><br>As with many problems, money can be answer and one way to make sure you leave enough to provide for your family, is with term life insurance. So, how much insurance is right for you and your family. Premium prices and the term of the policy can vary dramatically. Life insurance companies may be hesitant to classify e-cigarettes as anything other than nicotine use, but that doesn't mean e-cigarette users don't have options when applying for life insurance. s important to keep updating your beneficiaries on your life insurance policies. Online calculator tools for estimating the cost of child rearing and other expenses which may need to be considered can be useful. Well, the purpose of this article is not to put down Primerica, nor is it to sing the praises of Primerica. <br><br>Further, some business owner life insurance premiums can be paid from your business and used as a tax deduction expense. Upon your death the full face amount will be paid to your loved ones, regardless of how you die other than by suicide. Insurance companies in the United States obey the Surgeon Generals Office of the United States and uphold "life", the American Medical Association, and the Food and Drug Administration. Whole life agents also claim you need life insurance for your whole life and whole life is the best option. Focus on 3 major motives of the coverage, length and cheap in mind, when assessing the best policy. The biggest disadvantage of a temporary medical insurance plan over an individual health insurance plan is that if you become sick or injured while on the temporary plan, and it expires while you are still receiving medical treatment, you will no longer have medical insurance to pay for your medical treatments, and you will not qualify to extend the temporary medical insurance plan nor will you qualify for individual health insurance plans. Here each year's annual renewable term rates are included with time value of money to decide on the premium level.  <br><br>Call us at 1300 366 817 for finding the best policy for your needs. When it comes to life insurance companies, famous names are all apart of the game, but do not let their name recognition lull you into a sense of security. As we mentioned above, if your health is not the best, you will not likely be approved for no exam life. Are you uncertain whether you want the life insurance to work as an investment for you. When constructing a solid financial base, term life insurance is always a good investment because of its low cost. I would like to suggest to you that the internet is the best place to get information when you are in search of life insurance. Premiums that cover adrenaline junkies can vary significantly between carriers.  <br><br>You could convert your term policy to a permanent policy at that time to guarantee that you always have coverage. If the events covered never happen you paid a premium without receiving any amount of your premium back. Death is not something most young people want to consider, but life insurance is really not about death. You see if you buy a 25 or 30 year term policy from the outset then all is taken care of. If you do that, you may be running the risk of getting what you paid for if it. The above reasons will definitely delay the payment of a claim while they decide to pay or not the claim. Keep in mind that at the insured, you are not the direct beneficiary because everything will go to your family. <br><br>If you will not need to make a [http://www.Reddit.com/r/howto/search?q=cash+withdrawal cash withdrawal] from your annuity, you can purchase an immediate annuity which skips the accumulation phase altogether. You can use a ‘quote and compare’ service to find the best rate for your [http://www.termlifepolicy.com/insurance-agents/kansas/ term life policy]. Really state-of-the-art computer software enables the web services to provide several quotes from numerous distinctive companies-all in a single place, all in only seconds. Life insurance can be used to protect from the loss of income if the breadwinner passes on. Take time to check our page if you want to know more about insurance. This affects the usage of the product as well as the limitations on the collection of the benefit. Our reputation in the market makes us proud of our achievements.  <br><br>After Reading About This Diet, Snake Oil is Actually Starting to Sound Really Good. Long term life insurance really blurs the line between whole life and term life insurance, with policies often borrowing from both structures to offer the customer even more flexibility. The main advantage of term life insurance over permanent life insurance policy is cost. Why your investment portion of the policy that's where. It is important to assess your situation and determine whether or not you even need life insurance coverage. Whole life is insurance along with some kind of investments. The carrier has to calculate the risk of passing away in year 20, year 19, year 18, and so on based on the attained age at that time.
 
: <math>{n\choose k}_2</math>.
 
Rows are counted starting from 0. The entries of the <math>n</math>-th row are indexed starting with <math>-n</math> from the left, and the middle entry has index 0. The symmetry of the entries of a row about the middle entry is expressed by the relationship
 
: <math>{n\choose k}_2={n\choose-k}_2</math>
 
== Properties ==
 
The <math>n</math>-th row corresponds to the coefficients in the [[polynomial expansion]] of the expansion of the [[trinomial]] <math>(1 + x + x^2)</math> raised to the <math>n</math>-th power:<ref name="MathWorldTrinomialCoefficient">{{MathWorld|TrinomialCoefficient|Trinominal Coefficient}}</ref>
 
:<math>\left(1+x+x^2\right)^n= \sum _{j=0}^{2n}{n\choose j-n}_2 x^{j}=\sum _{k=-n}^{n}{n\choose k}_2 x^{n+k}</math>
 
or, symmetrically,
:<math>\left(1+x+1/x\right)^n=\sum_{k=-n}^{n}{n\choose k}_2 x^k</math>,
hence the alternative name '''trinomial coefficients''' because of their relationship to the [[multinomial coefficients]]:
 
: <math>{n\choose k}_2=\sum_{\textstyle{0\leq\mu,\nu\leq n\atop\mu+2\nu=n+k}}\frac{n!}{\mu!\,\nu!\,(n-\mu-\nu)!}</math>
 
Furthermore, the diagonals have interesting properties, such as their relationship to the [[triangular numbers]].
 
The sum of the elements of <math>n</math>-th row is <math>3^n</math>.
 
== Recursion formula ==
 
The trinomial coefficients can be generated using the following [[Recursion|recursion formula]]:<ref name="MathWorldTrinomialCoefficient"/>
 
:<math>{0\choose 0}_2=1</math>,
 
:<math>{n+1\choose k}_2={n\choose k-1}_2+{n\choose k}_2+{n\choose k+1}_2</math> for <math>n\geq 0</math>,
 
where <math>{n\choose k}_2=0</math> for <math>\ k<-n</math> and <math>\ k>n</math>.
 
== The middle entries ==
 
The middle entries of the trinomial triangle {{OEIS|A002426}}
 
: 1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, …
 
were studied by [[Leonhard Euler|Euler]]. The middle entry for the <math>n</math>-th row is given by
 
: <math>{n\choose0}_2=\sum_{k=0}^n\frac{n(n-1)\cdots(n-2k+1)}{(k!)^2}=\sum_{k=0}^n{n\choose 2k}{2k\choose k}.</math>
 
The corresponding [[generating function]] is<ref>{{MathWorld|CentralTrinomialCoefficient|Central Trinomial Coefficient}}</ref>
 
: <math>1+x+3x^2+7x^3+19x^4+\ldots=\frac1{\sqrt{(1+x)(1-3x)}}.</math>
 
Euler also noted the following ''exemplum memorabile inductionis fallacis'' ("notable example of fallacious induction"):
 
: <math>3{n+1\choose0}_2-{n+2\choose0}_2=f_n(f_n+1)</math> for <math>0\leq n\leq 7</math>
 
where <math>f_n</math> stands for the [[Fibonacci sequence]]. For larger <math>n</math>, however, this relationship is incorrect. [[George Andrews (mathematician)|George Andrews]] explained this fallacy using the general identity<ref>George Andrews, Three Aspects for Partitions. ''Séminaire Lotharingien de Combinatoire'', B25f (1990) [http://www.mat.univie.ac.at/~slc/opapers/s25andrews.html Online copy]</ref>
   
: <math>2\sum_{k\in\mathbb Z}\left[{n+1\choose 10k}_2-{n+1\choose 10k+1}_2\right]=f_n(f_n+1).</math>
 
== Chess mathematics ==
 
[[File:King walks.svg|thumb|right|Number of ways to reach a cell with the minimum number of moves]]
 
The triangle corresponds to the number of possible paths that can be taken by the [[King (chess)|king]] in a game of [[chess]]. The entry in a cell represents the number of different paths (using a minimum number of moves) the king can take to reach the cell.
 
<br style="clear:both;" />
 
== Importance in combinatorics ==
The coefficient of <math>x^k</math> in the polynomial expansion of <math>\left(1+x+x^2\right)^n</math> specifies the number of different ways of randomly drawing <math>k</math> cards from two sets of <math>n</math> identical playing cards.<ref name="ct200510">Andreas Stiller: ''Pärchenmathematik. Trinomiale und Doppelkopf.'' ("Pair mathematics.  Trinomials and the game of ''Doppelkopf''"). [[c't]] Heft 10/2005, p. 181ff</ref> For example, in such a card game with two sets of the three cards A, B, C, the choices look like this:
 
{| class="prettytable"
!bgcolor=ececec|Number of selected cards
!bgcolor=ececec|Number of options
!bgcolor=ececec|Options
|-
|0
|1
|
|-
|1
|3
|A, B, C
|-
|2
|6
|AA, AB, AC, BB, BC, CC
|-
|3
|7
|AAB, AAC, ABB, ABC, ACC, BBC, BCC
|-
|4
|6
|AABB, AABC, AACC, ABBC, ABCC, BBCC
|-
|5
|3
|AABBC, AABCC, ABBCC
|-
|6
|1
|AABBCC
|}
 
In particular, this results in <math>{24\choose 12-24}_2={24\choose -12}_2={24\choose 12}_2</math> as the number of different hands in a game of ''[[Doppelkopf]]''.
 
Alternatively, it is also possible to arrive at this number by considering the number of ways of choosing <math>p</math> pairs of identical cards from the two sets, which is <math>{n\choose p}</math>. The remaining <math>k-2p</math> cards can then be chosen in <math>{n-p\choose k-2p}</math> ways,<ref name="ct200510"/> which can be written in terms of the [[binomial coefficients]] as
 
:<math>{n\choose k-n}_2=\sum_{p=\max(0,k-n)}^{\min(n,[k/2])}{n\choose p}{n-p \choose k-2p}</math>.
 
For example,
 
:<math>6={3\choose 2-3}_2={3\choose 0}{3\choose 2}+{3\choose 1}{2\choose 0}=1\cdot 3+3\cdot 1</math>.
 
The example above corresponds to the three ways of selecting two cards without pairs of identical cards (AB, AC, BC) and the three ways of selecting a pair of identical cards (AA, BB, CC).
 
== References ==
{{reflist}}
 
== Further reading ==
 
* [[Leonhard Euler]], ''Observationes analyticae'' ("Analytical observations"). ''Novi Commentarii academiae scientiarum Petropolitanae'' 11 (1767) 124–143 [http://www.math.dartmouth.edu/~euler/pages/E326.html PDF]
 
[[Category:Discrete mathematics]]
[[Category:Triangles of numbers]]
[[Category:Factorial and binomial topics]]

Latest revision as of 23:00, 11 August 2014


Visiting at least three quotes sites raise the chances that you would realize more savings. On the one hand, you have people who are strong and mighty. But guaranteed issue policies usually refund all premiums paid with interest even if the waiting period has not been satisfied. Look for the carriers that are rated A+ or better by the A. A lot of agencies out there are going to make some determined sales phone calls to arrange an appointment with you. In the unthinkable event of your passing, the right life insurance coverage will be absolutely crucial to your family's future well-being. You can do it all online, and there is no need of an agent.

As with many problems, money can be answer and one way to make sure you leave enough to provide for your family, is with term life insurance. So, how much insurance is right for you and your family. Premium prices and the term of the policy can vary dramatically. Life insurance companies may be hesitant to classify e-cigarettes as anything other than nicotine use, but that doesn't mean e-cigarette users don't have options when applying for life insurance. s important to keep updating your beneficiaries on your life insurance policies. Online calculator tools for estimating the cost of child rearing and other expenses which may need to be considered can be useful. Well, the purpose of this article is not to put down Primerica, nor is it to sing the praises of Primerica.

Further, some business owner life insurance premiums can be paid from your business and used as a tax deduction expense. Upon your death the full face amount will be paid to your loved ones, regardless of how you die other than by suicide. Insurance companies in the United States obey the Surgeon Generals Office of the United States and uphold "life", the American Medical Association, and the Food and Drug Administration. Whole life agents also claim you need life insurance for your whole life and whole life is the best option. Focus on 3 major motives of the coverage, length and cheap in mind, when assessing the best policy. The biggest disadvantage of a temporary medical insurance plan over an individual health insurance plan is that if you become sick or injured while on the temporary plan, and it expires while you are still receiving medical treatment, you will no longer have medical insurance to pay for your medical treatments, and you will not qualify to extend the temporary medical insurance plan nor will you qualify for individual health insurance plans. Here each year's annual renewable term rates are included with time value of money to decide on the premium level.

Call us at 1300 366 817 for finding the best policy for your needs. When it comes to life insurance companies, famous names are all apart of the game, but do not let their name recognition lull you into a sense of security. As we mentioned above, if your health is not the best, you will not likely be approved for no exam life. Are you uncertain whether you want the life insurance to work as an investment for you. When constructing a solid financial base, term life insurance is always a good investment because of its low cost. I would like to suggest to you that the internet is the best place to get information when you are in search of life insurance. Premiums that cover adrenaline junkies can vary significantly between carriers.

You could convert your term policy to a permanent policy at that time to guarantee that you always have coverage. If the events covered never happen you paid a premium without receiving any amount of your premium back. Death is not something most young people want to consider, but life insurance is really not about death. You see if you buy a 25 or 30 year term policy from the outset then all is taken care of. If you do that, you may be running the risk of getting what you paid for if it. The above reasons will definitely delay the payment of a claim while they decide to pay or not the claim. Keep in mind that at the insured, you are not the direct beneficiary because everything will go to your family.

If you will not need to make a cash withdrawal from your annuity, you can purchase an immediate annuity which skips the accumulation phase altogether. You can use a ‘quote and compare’ service to find the best rate for your term life policy. Really state-of-the-art computer software enables the web services to provide several quotes from numerous distinctive companies-all in a single place, all in only seconds. Life insurance can be used to protect from the loss of income if the breadwinner passes on. Take time to check our page if you want to know more about insurance. This affects the usage of the product as well as the limitations on the collection of the benefit. Our reputation in the market makes us proud of our achievements.

After Reading About This Diet, Snake Oil is Actually Starting to Sound Really Good. Long term life insurance really blurs the line between whole life and term life insurance, with policies often borrowing from both structures to offer the customer even more flexibility. The main advantage of term life insurance over permanent life insurance policy is cost. Why your investment portion of the policy that's where. It is important to assess your situation and determine whether or not you even need life insurance coverage. Whole life is insurance along with some kind of investments. The carrier has to calculate the risk of passing away in year 20, year 19, year 18, and so on based on the attained age at that time.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Divine_coincidence&oldid=277658"