A174845 - OEIS

%I #40 Aug 10 2018 18:09:33

%S 1,1,8,153,4981,236970,15211158,1250791640,127078235560,

%T 15531504729378,2237017556966100,373533515381767037,

%U 71351421971134445583,15419725101750288678775,3734978285744386546427032,1005908662614385539285407741,299140901286981469075716747245

%N O.g.f.: Sum_{n>=0}  n^(2*n) * x^n / (1 - n^2*x)^n * exp( -n^2*x / (1 - n^2*x) ) / n!.

%H G. C. Greubel, <a href="/A174845/b174845.txt">Table of n, a(n) for n = 0..268</a>

%F a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(2*n,k) for n>0 with a(0)=1. - _Paul D. Hanna_, Mar 08 2013

%e O.g.f.: A(x) = 1 + x + 8*x^2 + 153*x^3 + 4981*x^4 + 236970*x^5 +...

%e where

%e A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^4*x^2/(1-2^2*x)^2*exp(-2^2*x/(1-2^2*x))/2! + 3^6*x^3/(1-3^2*x)^3*exp(-3^2*x/(1-3^2*x))/3! + 4^8*x^4/(1-4^2*x)^4*exp(-4^2*x/(1-4^2*x))/4! +...

%e simplifies to a power series in x with integer coefficients.

%t Flatten[{1,Table[Sum[Binomial[n-1,k-1] * StirlingS2[2*n,k],{k,1,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Aug 11 2014 *)

%t a[ n_] := SeriesCoefficient[ 1 + Sum[(k^2 x)^k / (1 - k^2 x)^k Exp[-k^2 x / (1 - k^2 x)] / k!, {k, n + 1}], {x, 0, n}]; (* _Michael Somos_, Jun 27 2017 *)

%o (PARI) a(n)=polcoeff(sum(k=0, n+1, (k^2*x)^k/(1-k^2*x)^k*exp(-k^2*x/(1-k^2*x+x*O(x^n)))/k!), n) \\ _Paul D. Hanna_, Nov 04 2012

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)

%o {a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(2*n, k)))}

%o for(n=0,25,print1(a(n),", "))\\ _Paul D. Hanna_, Mar 08 2013

%Y Cf. A134055, A222053, A222054, A243942, A008277.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 06 2012