A227851 G.f.: Sum_{n>=1} n^n * x^n * Product_{k=1..n} (k + x) / (1 + n*k*x + n*x^2).

1, 1, 8, 125, 3650, 171440, 11815940, 1122759980, 140645621840, 22456283261240, 4451225265169640, 1072410309912462440, 308628265617560695880, 104567048162852196877640, 41198829781936190483346440, 18676924223093561435394148040, 9652952812685808726911849225480

Offset: 0

Author

Paul D. Hanna, Oct 31 2013

Keywords

• nonn

Comments

Compare to the identity:

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

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 125*x^3 + 3650*x^4 + 171440*x^5 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2^2*x^2*(1+x)*(2+x)/((1+2*x+2*x^2)*(1+4*x+2*x^2)) + 3^3*x^3*(1+x)*(2+x)*(3+x)/((1+3*x+3*x^2)*(1+6*x+3*x^2)*(1+9*x+3*x^2)) + 4^4*x^4*(1+x)*(2+x)*(3+x)*(4+x)/((1+4*x+4*x^2)*(1+8*x+4*x^2)*(1+12*x+4*x^2)*(1+16*x+4*x^2)) +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..236

Crossrefs

Cf. A204064.

Programs

PARI

{a(n)=polcoeff( sum(m=0, n, m^m*x^m*prod(k=1, m, (k+x)/(1+m*k*x+m*x^2 +x*O(x^n))) ), n)}
for(n=0, 20, print1(a(n), ", "))

Formula

a(n) == 2 (mod 3) for n>1.

a(n) == 0 (mod 5) for n>2.

a(n) == 0 (mod 8) for n>7.

a(n) ~ sqrt(Pi) * n^(2*n+1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Nov 03 2014