A218687 O.g.f.: Sum_{n>=0} n^n * (1+n^3*x)^n * x^n/n! * exp(-n*(1+n^3*x)*x).

1, 1, 2, 31, 398, 10476, 296407, 12613297, 592445192, 36797742660, 2524966492661, 212912151736648, 19819138754732997, 2155966497948737905, 259256365067737582615, 35050667748654756208069, 5257919606219599751747894, 858816581875175776426876930

Offset: 0

Author

Paul D. Hanna, Nov 05 2012

Keywords

• nonn

Comments

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

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

Examples

			O.g.f: A(x) = 1 + x + 2*x^2 + 31*x^3 + 398*x^4 + 10476*x^5 + 296407*x^6 +...
where
A(x) = 1 + (1+x)*x*exp(-(1+x)*x) + 2^2*(1+2^3*x)^2*x^2/2!*exp(-2*(1+2^3*x)*x) + 3^3*(1+3^3*x)^3*x^3/3!*exp(-3*(1+3^3*x)*x) + 4^4*(1+4^3*x)^4*x^4/4!*exp(-4*(1+4^3*x)*x) + 5^5*(1+5^3*x)^5*x^5/5!*exp(-5*(1+5^3*x)*x) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218685, A218686, A218684.

Programs

PARI

{a(n)=polcoeff(sum(k=0,n,k^k*(1+k^3*x)^k*x^k/k!*exp(-k*x*(1+k^3*x)+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))