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

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

Offset: 0

Author

Paul D. Hanna, Nov 20 2011

Keywords

• nonn

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.