A218677 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^(2*n) * x^n/n! * exp(-n*x*(1+n*x)^2).

1, 1, 3, 14, 79, 516, 3802, 30668, 268815, 2522594, 25201736, 266014607, 2953171684, 34326755191, 416313253084, 5251970372080, 68737673434847, 931207966502919, 13031639620371226, 188051624603419973, 2793741995189126920, 42668132798523737471, 669061042470049870917

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 + 3*x^2 + 14*x^3 + 79*x^4 + 516*x^5 + 3802*x^6 +...
where
A(x) = 1 + (1+x)^2*x*exp(-x*(1+x)^2) + 2^2*(1+2*x)^4*x^2/2!*exp(-2*x*(1+2*x)^2) + 3^3*(1+3*x)^6*x^3/3!*exp(-3*x*(1+3*x)^2) + 4^4*(1+4*x)^8*x^4/4!*exp(-4*x*(1+4*x)^2) + 5^5*(1+5*x)^10*x^5/5!*exp(-5*x*(1+5*x)^2) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218670, A218678, A218679.

Programs

PARI

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