A135746 E.g.f.: A(x) = Sum_{n>=0} exp(n^2*x) * x^n/n!.

1, 1, 3, 16, 137, 1536, 22417, 407884, 8920641, 230576320, 6928080641, 238375169484, 9288784476193, 406150114297552, 19761959813464065, 1062437048084297596, 62727815353861478273, 4045278841893314992896

Offset

0,3

Links

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

Formula

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

O.g.f.: Sum_{n>=0} x^n/(1 - n^2*x)^(n+1). - Paul D. Hanna, Aug 08 2009

a(n) ~ n^(n + 1/2) * r^(2*n - 3*r + 1/2) / (sqrt(2*n + 3*r) * (n - r)^(n - r + 1/2)), where r = (n/w) * (1 + (w-1)/((2*w^2 + w - 2)/log(w-1) - w + 2)) and w = LambertW(exp(1)*n). - Vaclav Kotesovec, Jul 05 2022

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1536*x^5/5! + ...
where A(x) = 1 + exp(x)*x + exp(4*x)*x^2/2! + exp(9*x)*x^3/3! + exp(16*x)*x^4/4! + exp(25*x)*x^5/5! + ...
O.g.f.: F(x) = 1 + x + 3*x^2 + 16*x^3 + 137*x^4 + 1536*x^5 + 22417*x^6 + ...
where F(x) = 1 + x/(1-x)^2 + x^2/(1-4*x)^3 + x^3/(1-9*x)^4 + x^4/(1-16*x)^5 + x^5/(1-25*x)^6 + ...

Mathematica

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

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*(k^2)^(n-k))}
(PARI) {a(n)=n!*polcoeff(sum(k=0, n, exp(k^2*x +x*O(x^n))*x^k/k!), n)}
(PARI) {a(n)=polcoeff(sum(k=0, n, x^k/(1-k^2*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Aug 08 2009

Crossrefs

Cf. variants: A135742, A135743, A135744, A135745, A135747.

Cf. A000248.

Sequence in context: A357088 A277673 A335356 * A368293 A345349 A230320

Adjacent sequences: A135743 A135744 A135745 * A135747 A135748 A135749

Keyword

nonn

Author

Paul D. Hanna, Nov 27 2007

Status

approved