|
%I #9 Nov 05 2016 13:13:05
%S 1,1,3,13,83,686,7132,90343,1357449,23783068,478784096,10938189329,
%T 280771780489,8029138915630,253911056912892,8823070442039641,
%U 335009138739028673,13830540214264709000,618085473234055115968
%N E.g.f.: A(x) = Sum_{n>=0} exp(n*(n+1)/2*x)*x^n/n!.
%H G. C. Greubel, <a href="/A135743/b135743.txt">Table of n, a(n) for n = 0..250</a>
%F a(n) = Sum_{k=0..n} C(n,k)*[k*(k+1)/2]^(n-k).
%F O.g.f.: Sum_{n>=0} x^n/(1 - n(n+1)/2*x)^(n+1).
%e E.g.f.: 1 + x + 3*x^2/2! + 13*x^3/3! + 83*x^4/4! +...
%e = 1 + exp(x)*x + exp(3x)*x^2/2! + exp(6x)*x^3/3! + exp(10x)*x^4/4! +...
%e O.g.f.: 1 + x + 3*x^2 + 13*x^3 + 83*x^4 + 686*x^5 +...
%e = 1 + x/(1-x)^2 + x^2/(1-3x)^3 + x^3/(1-6x)^4 + x^4/(1-10x)^5 +...
%t 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 *)
%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)*(k*(k+1)/2)^(n-k))}
%o (PARI) {a(n)=n!*polcoeff(sum(k=0,n,exp(k*(k+1)/2*x +x*O(x^n))*x^k/k!),n)}
%o (PARI) {a(n)=polcoeff(sum(k=0, n, x^k/(1-k*(k+1)/2*x +x*O(x^n))^(k+1)), n)}
%Y Cf. variants: A135742, A135744, A135745, A135746.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 27 2007
|
