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%I #52 Jan 20 2021 05:50:05
%S 1,1,6,65,1050,22827,627396,20912320,820784250,37112163803,
%T 1900842429486,108823356051137,6888836057922000,477898618396288260,
%U 36060660300744309600,2940812098256837097720,257780560811305783833450,24171700822696604400643035,2414448376056191692970387250
%N O.g.f.: Sum_{n>=1} (n^2)^(n-1) * exp(-n^2*x) * x^n / n!.
%C For n>1, a(n) is the number of set partitions of [2*n-2] into n blocks, i.e., Stirling2(2*n-2, n). E.g., a(3) = 6: [12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34]. - _Yuchun Ji_, Jan 12 2021
%H G. C. Greubel, <a href="/A217899/b217899.txt">Table of n, a(n) for n = 1..250</a>
%F a(n) = (1/n!) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * (k^2)^(n-1).
%F a(n) = [x^n] x + x^2/Product_{k=1..n} (1-k*x).
%F a(n) = [x^n] x + x^2*(1+x)^(2*n-3) / Product_{k=1..n-1} (1-k*x).
%F a(n) = Sum_{j=0..n-1} binomial(2*n-1,j)*Stirling2(2*n-j-1,n). - _Vladimir Kruchinin_, Jun 14 2013
%F a(n) ~ 2^(2*n-5/2) * n^(n-5/2) / (sqrt(Pi*(1-c)) * exp(n) * c^n *(2-c)^(n-2)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, May 20 2014
%e O.g.f.: A(x) = x + x^2 + 6*x^3 + 65*x^4 + 1050*x^5 + 22827*x^6 + 627396*x^7 + ... where A(x) = 1^0*x*exp(-1*x) + 2^2*exp(-2^2*x)*x^2/2! + 3^4*exp(-3^2*x)*x^3/3! + 4^6*exp(-4^2*x)*x^4/4! + 5^8*exp(-5^2*x)*x^5/5! + ... simplifies to a power series in x with integer coefficients.
%t a[n_] := Sum[ Binomial[2*n - 3, j]*StirlingS2[2*n - j - 3, n-1], {j, 0, n-2}]; a[1] = 1; Table[a[n], {n, 1, 19}] (* _Jean-François Alcover_, Jun 14 2013, after _Vladimir Kruchinin_ *)
%o (PARI) {a(n)=polcoeff(sum(m=1,n,(m^2)^(m-1)*x^m*exp(-m^2*x+x*O(x^n))/m!),n)}
%o (PARI) {a(n)=1/n!*sum(k=1,n, (-1)^(n-k)*binomial(n,k)*(k^2)^(n-1))}
%o (PARI) {a(n)=polcoeff(x+x^2/prod(k=1, n, 1-k*x +x*O(x^n)), n)}
%o (PARI) {a(n)=polcoeff(x+x^2*(1+x)^(2*n-3)/prod(k=0, n-1, 1-k*x +x*O(x^n)), n)}
%o for(n=1,20,print1(a(n),", "))
%Y Cf. A217900, A217901, A217902, A217903, A217904, A217905.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Oct 14 2012
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