A218684 O.g.f.: Sum_{n>=0} (1+n^2*x)^n * x^n/n! * exp(-(1+n^2*x)*x).

1, 0, 1, 2, 7, 18, 96, 260, 1851, 5270, 46515, 137942, 1447202, 4433772, 53787706, 169169912, 2326986783, 7477418982, 114916173009, 375898894514, 6380455164161, 21185872231238, 393499602818322, 1323362744628080, 26691270481453228, 90755667374332324

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 + 2*x^3 + 7*x^4 + 18*x^5 + 96*x^6 + 260*x^7 +...
where
A(x) = exp(-x) + (1+x)*x*exp(-(1+x)*x) + (1+2^2*x)^2*x^2/2!*exp(-(1+2^2*x)*x) + (1+3^2*x)^3*x^3/3!*exp(-(1+3^2*x)*x) + (1+4^2*x)^4*x^4/4!*exp(-(1+4^2*x)*x) + (1+5^2*x)^5*x^5/5!*exp(-(1+5^2*x)*x) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218686, A218685, A218687, A185040.

Programs

PARI

{a(n)=polcoeff(sum(k=0,n,(1+k^2*x)^k*x^k/k!*exp(-x*(1+k^2*x)+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))