A218678 O.g.f.: Sum_{n>=0} n^n * (1+n*x)^(3*n) * x^n/n! * exp(-n*x*(1+n*x)^3).

1, 1, 4, 22, 161, 1321, 12541, 130383, 1482875, 18153076, 237430711, 3295833146, 48274094584, 742868875984, 11963384310515, 200974595790271, 3511980095379727, 63682377891348689, 1195661594431548085, 23199930176668566579, 464421513762097397125, 9576744471125816269165

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 + 4*x^2 + 22*x^3 + 161*x^4 + 1321*x^5 + 12541*x^6 +...
where
A(x) = 1 + (1+x)^3*x*exp(-x*(1+x)^3) + 2^2*(1+2*x)^6*x^2/2!*exp(-2*x*(1+2*x)^3) + 3^3*(1+3*x)^9*x^3/3!*exp(-3*x*(1+3*x)^3) + 4^4*(1+4*x)^12*x^4/4!*exp(-4*x*(1+4*x)^3) + 5^5*(1+5*x)^15*x^5/5!*exp(-5*x*(1+5*x)^3) +...
simplifies to a power series in x with integer coefficients.
		

Crossrefs

Cf. A218670, A218677, A218679.

Programs

PARI

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