A135746 - OEIS

%I #18 Jul 05 2022 04:45:16

%S 1,1,3,16,137,1536,22417,407884,8920641,230576320,6928080641,

%T 238375169484,9288784476193,406150114297552,19761959813464065,

%U 1062437048084297596,62727815353861478273,4045278841893314992896

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

%H G. C. Greubel, <a href="/A135746/b135746.txt">Table of n, a(n) for n = 0..250</a>

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

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

%F 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

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1536*x^5/5! + ...

%e 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! + ...

%e O.g.f.: F(x) = 1 + x + 3*x^2 + 16*x^3 + 137*x^4 + 1536*x^5 + 22417*x^6 + ...

%e 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 + ...

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

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

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

%o (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

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

%Y Cf. A000248.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 27 2007