A174845 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!.

1, 1, 8, 153, 4981, 236970, 15211158, 1250791640, 127078235560, 15531504729378, 2237017556966100, 373533515381767037, 71351421971134445583, 15419725101750288678775, 3734978285744386546427032, 1005908662614385539285407741, 299140901286981469075716747245

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..268

Formula

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

Example

O.g.f.: A(x) = 1 + x + 8*x^2 + 153*x^3 + 4981*x^4 + 236970*x^5 +...
where
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! +...
simplifies to a power series in x with integer coefficients.

Mathematica

Flatten[{1, Table[Sum[Binomial[n-1, k-1] * StirlingS2[2*n, k], {k, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)
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 *)

Prog

(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
for(n=0, 25, print1(a(n), ", "))
(PARI) Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)
{a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1) * Stirling2(2*n, k)))}
for(n=0, 25, print1(a(n), ", "))\\ Paul D. Hanna, Mar 08 2013

Crossrefs

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

Sequence in context: A247538 A360575 A304400 * A352253 A302259 A113268

Adjacent sequences: A174842 A174843 A174844 * A174846 A174847 A174848

Keyword

nonn

Author

Paul D. Hanna, Nov 06 2012

Status

approved