A229045 O.g.f.: Sum_{n>=0} x^n * Product_{k=1..n} (n*k + x) / (1 + n*k*x).

1, 1, 8, 120, 3516, 166328, 11532458, 1100805591, 138367161579, 22150155031528, 4399449078063293, 1061636331152735214, 305921694251090263429, 103759120501255220937827, 40916086296549562038319264, 18562249570081891918836678474, 9599588267110033395243891612720

Offset

0,3

Comments

Compare to a g.f. of Fibonacci numbers (A000045):

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Formula

a(n) ~ sqrt(Pi) * n^(2*n+1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Oct 28 2014

Example

G.f.: A(x) = 1 + x + 8*x^2 + 120*x^3 + 3516*x^4 + 166328*x^5 +...
where
A(x) = 1 + x*(1+x)/(1+x) + x^2*(2+x)*(4+x)/((1+2*x)*(1+4*x)) + x^3*(3+x)*(6+x)*(9+x)/((1+3*x)*(1+6*x)*(1+9*x)) + x^4*(4+x)*(8+x)*(12+x)*(16+x)/((1+4*x)*(1+8*x)*(1+12*x)*(1+16*x)) +...

Prog

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

Crossrefs

Cf. A220181, A230682.

Sequence in context: A045754 A339201 A360482 * A173774 A034669 A000848

Adjacent sequences: A229042 A229043 A229044 * A229046 A229047 A229048

Keyword

nonn

Author

Paul D. Hanna, Nov 02 2013

Status

approved