A135743 E.g.f.: A(x) = Sum_{n>=0} exp(n*(n+1)/2*x)*x^n/n!.

1, 1, 3, 13, 83, 686, 7132, 90343, 1357449, 23783068, 478784096, 10938189329, 280771780489, 8029138915630, 253911056912892, 8823070442039641, 335009138739028673, 13830540214264709000, 618085473234055115968

Offset: 0

Author

Paul D. Hanna, Nov 27 2007

Keywords

• nonn

Examples

			E.g.f.: 1 + x + 3*x^2/2! + 13*x^3/3! + 83*x^4/4! +...
= 1 + exp(x)*x + exp(3x)*x^2/2! + exp(6x)*x^3/3! + exp(10x)*x^4/4! +...
O.g.f.: 1 + x + 3*x^2 + 13*x^3 + 83*x^4 + 686*x^5 +...
= 1 + x/(1-x)^2 + x^2/(1-3x)^3 + x^3/(1-6x)^4 + x^4/(1-10x)^5 +...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. variants: A135742, A135744, A135745, A135746.

Programs

Mathematica

Flatten[{1, Table[Sum[Binomial[n, k]*Binomial[k + 1, 2]^(n - k), {k, 0, n}], {n, 1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)

PARI

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

PARI

{a(n)=n!*polcoeff(sum(k=0,n,exp(k*(k+1)/2*x +x*O(x^n))*x^k/k!),n)}

PARI

{a(n)=polcoeff(sum(k=0, n, x^k/(1-k*(k+1)/2*x +x*O(x^n))^(k+1)), n)}

Formula

a(n) = Sum_{k=0..n} C(n,k)*[k*(k+1)/2]^(n-k).

O.g.f.: Sum_{n>=0} x^n/(1 - n(n+1)/2*x)^(n+1).