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%I #9 Nov 05 2016 12:51:07
%S 1,1,1,4,19,131,1156,12622,166825,2600677,47038456,974165336,
%T 22829939089,599668759483,17512623094240,564613124026876,
%U 19972670155565761,771019774737952313,32326390781950804048
%N E.g.f.: A(x) = Sum_{n>=0} exp( n*(n-1)/2 * x ) * x^n / n!.
%H G. C. Greubel, <a href="/A135742/b135742.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). - _Paul D. Hanna_, Jul 30 2014
%t Flatten[{1, Table[Sum[Binomial[n, k]*Binomial[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*(k-1)/2)^(n-k))}
%o for(n=0,25,print1(a(n),", "))
%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 for(n=0,25,print1(a(n),", "))
%o (PARI) /* From Sum_{n>=0} x^n/(1 - n*(n-1)/2*x)^(n+1): */
%o {a(n)=polcoeff(sum(k=0, n, x^k/(1-k*(k-1)/2*x +x*O(x^n))^(k+1)), n)}
%o for(n=0,25,print1(a(n),", "))
%Y Cf. variants: A135743, A135744, A135745, A135746.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Nov 27 2007
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