A196457 - OEIS

%I #12 Jul 13 2019 23:46:50

%S 1,3,31,729,96895,35927793,81108563671,567783612614529,

%T 19581520178825073535,2420011073132910603900513,

%U 1292280969200128366004695992151,2658679109878459106807828064662797809,22431208469091982323298987880694649428158815,748294346623782293365235855701111498805828889778353

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

%C GENERAL BINOMIAL IDENTITY.

%C When p=-1, q=2, this sequence illustrates the following identity.

%C 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!,

%C 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;

%C which is a special case of the more general binomial identity:

%C 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.

%F GENERATING FUNCTIONS.

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

%F 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

%F FORMULAS FOR TERMS.

%F a(n) = Sum_{k=0..n} C(n,k)*(2^k + (-1)^k)^n.

%e E.g.f.: A(x) = 1 + 3*x + 31*x^2/2! + 729*x^3/3! + 96895*x^4/4! +...

%e where

%e 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! +...

%e or, equivalently,

%e 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! +...

%e Illustrate the formula for the terms:

%e a(1) = (1+1) + (2-1) = 3 ;

%e a(2) = (1+1)^2 + 2*(2-1)^2 + (2^2+1)^2 = 2^2 + 2*1^2 + 5^2 = 31 ;

%e 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 ;

%e 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.

%o (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)}

%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)*(2^k + (-1)^k)^n)}

%o (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)}

%o (PARI) {a(n)=local(p=-1, q=2, s=1, t=1, u=1, v=1);

%o sum(k=0, n, binomial(n, k)*(s*p^k + t*q^k)^(n-k)*(u*p^k + v*q^k)^k)}

%o (PARI) /* right side of the general binomial identity: */

%o {a(n)=local(p=-1, q=2, s=1, t=1, u=1, v=1);

%o sum(k=0, n, binomial(n, k)*(s + u*p^(n-k)*q^k)^(n-k) * (t + v*p^(n-k)*q^k)^k)}

%Y Cf. A138247, A196458, A196459, A196460.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 03 2011