A218679 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^(4*n) * x^n/n! * exp(-n*x*(1+n*x)^4).

1, 1, 5, 31, 273, 2652, 30071, 375628, 5135649, 75945388, 1202006514, 20243446719, 360517872287, 6758311053521, 132833835618576, 2728019848249377, 58370987166092073, 1297916560174624569, 29924140267551540116, 713934350929955200551, 17594768127940813003452

Offset: 0

Author

Paul D. Hanna, Nov 04 2012

Keywords

• nonn

Comments

Compare o.g.f. to the curious identity:

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

Examples

			O.g.f.: A(x) = 1 + x + 5*x^2 + 31*x^3 + 273*x^4 + 2652*x^5 + 30071*x^6 +...
where
A(x) = 1 + (1+x)^4*x*exp(-x*(1+x)^4) + 2^2*(1+2*x)^8*x^2/2!*exp(-2*x*(1+2*x)^4) + 3^3*(1+3*x)^12*x^3/3!*exp(-3*x*(1+3*x)^4) + 4^4*(1+4*x)^16*x^4/4!*exp(-4*x*(1+4*x)^4) + 5^5*(1+5*x)^20*x^5/5!*exp(-5*x*(1+5*x)^4) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218670, A218677, A218678.

Programs

PARI

{a(n)=local(A=1+x);A=sum(k=0,n,k^k*(1+k*x)^(4*k)*x^k/k!*exp(-k*x*(1+k*x)^4+x*O(x^n)));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))