A218685 O.g.f.: Sum_{n>=0} (1+n^3*x)^n * x^n/n! * exp(-(1+n^3*x)*x).

1, 0, 1, 6, 34, 270, 3415, 31230, 681026, 6949920, 230637870, 2546120514, 119281951006, 1394371349490, 87612425583018, 1069010047029672, 86763885548985810, 1094149501538197236, 111443560982774811439, 1442387644419293694144, 180179254059921915232864

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 + 6*x^3 + 34*x^4 + 270*x^5 + 3415*x^6 +...
where
A(x) = exp(-x) + (1+x)*x*exp(-(1+x)*x) + (1+2^3*x)^2*x^2/2!*exp(-(1+2^3*x)*x) + (1+3^3*x)^3*x^3/3!*exp(-(1+3^3*x)*x) + (1+4^3*x)^4*x^4/4!*exp(-(1+4^3*x)*x) + (1+5^3*x)^5*x^5/5!*exp(-(1+5^3*x)*x) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218687, A218684, A218686.

Programs

PARI

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