A245109 G.f.: Sum_{n>=0} exp(-(1 + n^2*x)) * (1 + n^2*x)^n / n!.

1, 3, 31, 520, 11991, 350889, 12428746, 516450792, 24619176153, 1323971052261, 79280864647205, 5231080689880500, 377062508515478306, 29479066783583059530, 2484534527715953700780, 224559818606249783480400, 21666961097367611148157815, 2222844864226101120054773295

Offset: 0

Author

Paul D. Hanna, Jul 12 2014

Keywords

• nonn

Comments

Compare the g.f. to:

(1) Sum_{n>=0} exp(-(1+n*x)) * (1+n*x)^n / n! = 1/(1-x).

(2) Sum_{n>=1} exp(-n^2*x) * n^(2*n) * x^n/n! = Sum_{n>=1} S2(2*n,n)*x^n (A007820).

Examples

			G.f.: A(x) = 1 + 3*x + 31*x^2 + 520*x^3 + 11991*x^4 + 350889*x^5 +...
where
A(x) = exp(-1) + exp(-(1+x))*(1+x) + exp(-(1+2^2*x))*(1+2^2*x)^2/2!
+ exp(-(1+3^2*x))*(1+3^2*x)^3/3! + exp(-(1+4^2*x))*(1+4^2*x)^4/4!
+ exp(-(1+5^2*x))*(1+5^2*x)^5/5! + exp(-(1+6^2*x))*(1+6^2*x)^6/6!
+ exp(-(1+7^2*x))*(1+7^2*x)^7/7! + exp(-(1+8^2*x))*(1+8^2*x)^8/8! +...
simplifies to a power series in x with integer coefficients.
		

Links

• Paul D. Hanna and Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (first 100 terms from Paul D. Hanna)

Crossrefs

Cf. A049020, A245110.

Cf. A187655, A217899, A217900.

Programs

Mathematica

Table[SeriesCoefficient[Sum[E^(-(1+k^2*x))*(1+k^2*x)^k/k!,{k,0,Infinity}],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 12 2014 *)

PARI

/* Must first set suitable precision */ \p300
{a(n)=local(A=1+x); A=suminf(k=0, exp(-(1+k^2*x)+x*O(x^n))*(1+k^2*x)^k/k!); round(polcoeff(A, n))}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) ~ c * d^n * (n-1)!, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348035823168..., and c = 10.427337127699040838035... . - Vaclav Kotesovec, Jul 12 2014

a(n) = A049020(2n,n). - Alois P. Heinz, Aug 23 2017