cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A138247 E.g.f.: Sum_{n>=0} exp( (2^n+3^n)*x ) * (2^n+3^n)^n * x^n/n!.

Original entry on oeis.org

1, 7, 223, 49849, 94705663, 1616229320497, 251286598125520183, 357716675257916544062689, 4670472774542449929397808845183, 559006854195449142958954163012808059617, 612171730457531439763516750114999086563829844663, 6118056385739077528636842573416061383741677666682643900049
Offset: 0

Views

Author

Paul D. Hanna, Mar 09 2008, revised Mar 11 2008

Keywords

Comments

GENERAL BINOMIAL IDENTITY.
When p=2, q=3, this sequence illustrates the following identity:
Sum_{k=0..n} C(n,k)*(p^k + q^k)^n =
Sum_{k=0..n} C(n,k)*(1 + p^(n-k)*q^k)^n
which is a special case of the more general binomial identity:
Sum_{k=0..n} C(n,k)*(s*p^k + t*q^k)^(n-k) * (u*p^k + v*q^k)^k =
Sum_{k=0..n} C(n,k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k.

Examples

			E.g.f.: A(x) = 1 + 7*x + 223*x^2/2! + 49849*x^3/3! + 94705663*x^4/4! + 1616229320497*x^5/5! + 251286598125520183*x^6/6! + 357716675257916544062689*x^7/7! + 4670472774542449929397808845183*x^8/8! + ...
such that
A(x) = exp(2*x) + (2+3)*exp((2+3)*x)*x + (2^2+3^2)^2*exp((2^2+3^2)*x)*x^2/2! + (2^3+3^3)^3*exp((2^3+3^3)*x)*x^3/3! + (2^4+3^4)^4*exp((2^4+3^4)*x)*x^4/4! + ...
ORDINARY GENERATING FUNCTION.
O.g.f.: B(x) = 1 + 7*x + 223*x^2 + 49849*x^3 + 94705663*x^4 + 1616229320497*x^5 + 251286598125520183*x^6 + 357716675257916544062689*x^7 + ...
such that
B(x) = 1/(1-2*x) + (2+3)*x/(1 - (2+3)*x)^2 + (2^2+3^2)^2*x^2/(1 - (2^2+3^2)*x)^3 + (2^3+3^3)^3*x^3/(1 - (2^3+3^3)*x)^4 + (2^4+3^4)^4*x^4/(1 - (2^4+3^4)*x)^5 + ...
ILLUSTRATION OF TERMS.
a(1) = 2 + 5 = 3 + 4 = 7 ;
a(2) = 2^2 + 2*5^2 + 13^2 = 5^2 + 2*7^2 + 10^2 = 223 ;
a(3) = 2^3 + 3*5^3 + 3*13^3 + 35^3 = 9^3 + 3*13^3 + 3*19^3 + 28^3 = 49849.

Crossrefs

Cf. A007689, A094617, A196457, A196458, A196459, A196460.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*(2^k + 3^k)^n, {k, 0, n}], {n, 0, 12}] (* Vaclav Kotesovec, Jul 14 2019 *)
  • PARI
    {a(n)=local(p=2,q=3,s=1,t=1,u=1,v=1);
sum(k=0,n,binomial(n,k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}
/* right side of the general binomial identity: */
{a(n)=local(p=2,q=3,s=1,t=1,u=1,v=1);
sum(k=0,n,binomial(n,k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k)}

Formula

E.g.f.: Sum_{n>=0} (2^n + 3^n)^n * exp( (2^n + 3^n)*x ) * x^n / n!.
O.g.f.: Sum_{n>=0} (2^n + 3^n)^n * x^n / (1 - (2^n + 3^n)*x)^(n+1). - Paul D. Hanna, Jul 13 2019
FORMULAS FOR TERMS.
a(n) = Sum_{k=0..n} C(n,k)*(2^k + 3^k)^n.
a(n) = Sum_{k=0..n} C(n,k)*(1 + 2^(n-k)*3^k)^n.
a(n) = Sum_{k=0..n} C(n,k)*A007689(k)^n.
a(n) = Sum_{k=0..n} C(n,k)*A094617(n,k)^n.
a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jul 14 2019

A196457 E.g.f.: A(x) = Sum_{n>=0} exp((2^n + (-1)^n)*x) * (2^n + (-1)^n)^n * x^n/n!.

Original entry on oeis.org

1, 3, 31, 729, 96895, 35927793, 81108563671, 567783612614529, 19581520178825073535, 2420011073132910603900513, 1292280969200128366004695992151, 2658679109878459106807828064662797809, 22431208469091982323298987880694649428158815, 748294346623782293365235855701111498805828889778353
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2011

Keywords

Comments

GENERAL BINOMIAL IDENTITY.
When p=-1, q=2, this sequence illustrates the following identity.
Given e.g.f.: Sum_{n>=0} (p^n+q^n)^n*exp((p^n+q^n)*x)*x^n/n! = Sum_{n>=0} a(n)*x^n/n!,
then a(n) = Sum_{k=0..n} C(n,k)*(p^k + q^k)^n = Sum_{k=0..n} C(n,k)*(1 + p^(n-k)*q^k)^n;
which is a special case of the more general binomial identity:
Sum_{k=0..n} C(n,k)*(s*p^k + t*q^k)^(n-k) * (u*p^k + v*q^k)^k = Sum_{k=0..n} C(n,k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k.

Examples

			E.g.f.: A(x) = 1 + 3*x + 31*x^2/2! + 729*x^3/3! + 96895*x^4/4! +...
where
A(x) = exp((1+1)*x) + (2-1)*exp((2-1)*x)*x + (2^2+1)^2*exp((2^2+1)*x)*x^2/2! + (2^3-1)^3*exp((2^3-1)*x)*x^3/3! +...
or, equivalently,
A(x) = exp(2*x) + 1*exp(1*x)*x + 5^2*exp(5*x)*x^2/2! + 7^3*exp(7*x)*x^3/3! + 17^4*exp(17*x)*x^4/4! + 31^5*exp(31*x)*x^5/5! +...
Illustrate the formula for the terms:
a(1) = (1+1) + (2-1) = 3 ;
a(2) = (1+1)^2 + 2*(2-1)^2 + (2^2+1)^2 = 2^2 + 2*1^2 + 5^2 = 31 ;
a(3) = (1+1)^3 + 3*(2-1)^3 + 3*(2^2+1)^3 + (2^3-1)^3 = 2^3 + 3*1^3 + 3*5^3 + 7^3 = 729 ;
a(4) = (1+1)^4 + 4*(2-1)^4 + 6*(2^2+1)^4 + 4*(2^3-1)^4 + (2^4+1)^4 = 2^4 + 4*1^4 + 6*5^4 + 4*7^4 + 17^4 = 96895.

Crossrefs

Cf. A138247, A196458, A196459, A196460.

Programs

  • PARI
    {a(n)=n!*polcoeff(sum(m=0,n,exp((2^m+(-1)^m+x*O(x^n))*x)*(2^m+(-1)^m)^m*x^m/m!),n)}
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)*(2^k + (-1)^k)^n)}
  • PARI
    {a(n)=local(p=-1, q=2);n!*polcoeff(sum(m=0,n,(p^m+q^m)^m*exp((p^m+q^m+x*O(x^n))*x)*x^m/m!),n)}
  • PARI
    {a(n)=local(p=-1, q=2, s=1, t=1, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}
  • PARI
    /* right side of the general binomial identity: */
{a(n)=local(p=-1, q=2, s=1, t=1, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k)}

Formula

GENERATING FUNCTIONS.
E.g.f.: Sum_{n>=0} (2^n + (-1)^n)^n * exp( (2^n + (-1)^n)*x ) * x^n/n!.
O.g.f.: Sum_{n>=0} (2^n + (-1)^n)^n * x^n / (1 - (2^n + (-1)^n)*x)^(n+1). - Paul D. Hanna, Jul 13 2019
FORMULAS FOR TERMS.
a(n) = Sum_{k=0..n} C(n,k)*(2^k + (-1)^k)^n.

A196458 E.g.f.: A(x) = Sum_{n>=0} (3^n + (-1)^n)^n * exp((3^n + (-1)^n)*x) * x^n/n!.

Original entry on oeis.org

1, 4, 112, 20608, 47100160, 848654393344, 152543949079048192, 239308785705492230176768, 3442046584639832610980531077120, 443426848780270385458655031044167696384, 515552048984399455145742768443316759297510080512
Offset: 0

Views

Author

Paul D. Hanna, Nov 20 2011

Keywords

Comments

When p=-1, q=3, this sequence illustrates the following identity.
Given e.g.f.: Sum_{n>=0} (p^n+q^n)^n*exp((p^n+q^n)*x)*x^n/n! = Sum_{n>=0} a(n)*x^n/n!,
then a(n) = Sum_{k=0..n} C(n,k)*(p^k + q^k)^n = Sum_{k=0..n} C(n,k)*(1 + p^(n-k)*q^k)^n;
which is a special case of the more general binomial identity:
Sum_{k=0..n} C(n,k)*(s*p^k + t*q^k)^(n-k) * (u*p^k + v*q^k)^k = Sum_{k=0..n} C(n,k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k.

Examples

			 E.g.f.: A(x) = 1 + 4*x + 112*x^2/2! + 20608*x^3/3! + 47100160*x^4/4! +...
where
_ A(x) = exp((1+1)*x) + (3-1)*exp((3-1)*x)*x + (3^2+1)^2*exp((3^2+1)*x)*x^2/2! + (3^3-1)^3*exp((3^3-1)*x)*x^3/3! +...
or, equivalently,
_ A(x) = exp(2*x) + 2*exp(2*x)*x + 10^2*exp(10*x)*x^2/2! + 26^3*exp(26*x)*x^3/3! + 82^4*exp(82*x)*x^4/4! + 242^5*exp(242*x)*x^5/5! +...
Illustrate the formula for the terms:
a(1) = (1+1) + (3-1) = 4 ;
a(2) = (1+1)^2 + 2*(3-1)^2 + (3^2+1)^2 = 2^2 + 2*2^2 + 10^2 = 112 ;
a(3) = (1+1)^3 + 3*(3-1)^3 + 3*(3^2+1)^3 + (3^3-1)^3 = 2^3 + 3*2^3 + 3*10^3 + 26^3 = 20608 ;
a(4) = (1+1)^4 + 4*(3-1)^4 + 6*(3^2+1)^4 + 4*(3^3-1)^4 + (3^4+1)^4 = 2^4 + 4*2^4 + 6*10^4 + 4*26^4 + 82^4 = 47100160.

Crossrefs

Cf. A196457, A196459, A196460, A138247.

Programs

  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)*(3^k + (-1)^k)^n)}
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)*(1 + (-1)^(n-k)*3^k)^n)}
  • PARI
    {a(n)=local(p=-1, q=3);n!*polcoeff(sum(m=0,n,(p^m+q^m)^m*exp((p^m+q^m+x*O(x^n))*x)*x^m/m!),n)}
  • PARI
    {a(n)=local(p=-1, q=3, s=1, t=1, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}
  • PARI
    /* right side of the general binomial identity: */
{a(n)=local(p=-1, q=3, s=1, t=1, u=1, v=1);
sum(k=0, n, binomial(n, k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k)}

Formula

a(n) = Sum_{k=0..n} C(n,k)*(3^k + (-1)^k)^n.
a(n) = Sum_{k=0..n} C(n,k)*(1 + (-1)^(n-k)*3^k)^n.
Showing 1-3 of 3 results.

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