A249593 G.f.: Product_{n>=1} 1/(1 - x^n/n^3) = Sum_{n>=0} a(n)*x^n/n!^3.

1, 1, 9, 251, 16496, 2083824, 453803984, 156304214576, 80272385155584, 58631012094472704, 58713787327403063808, 78225670182020153384448, 135277046518915274471718912, 297374407080303931562525442048, 816367902369725640298981464096768

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..180

Formula

a(n) = Sum_{k=1..n} n!^2*(n-1)!/(n-k)!^3 * b(k) * a(n-k), where b(k) = Sum_{d|k} d^(1-3*k/d) and a(0) = 1 (after Vladeta Jovovic in A007841).

a(n) ~ c * n!^3, where c = Product_{k>=2} 1/(1-1/k^3) = 3*Pi/cosh(sqrt(3)*Pi/2) = 1.235488267746513477155075624616837... . - Vaclav Kotesovec, Mar 05 2016

Example

G.f.: A(x) = 1 + x + 9*x^2/2!^3 + 251*x^3/3!^3 + 16496*x^4/4!^3 +...
where
A(x) = 1/((1-x)*(1-x^2/2^3)*(1-x^3/3^3)*(1-x^4/4^3)*(1-x^5/5^3)*...).

Mathematica

Table[n!^3 * SeriesCoefficient[Product[1/(1 - x^m/m^3), {m, 1, n}], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 05 2016 *)

Prog

(PARI) {a(n)=n!^3*polcoeff(prod(k=1, n, 1/(1-x^k/k^3 +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Using logarithmic derivative: */
{b(k) = sumdiv(k, d, d^(1-3*k/d))}
{a(n) = if(n==0, 1, sum(k=1, n, n!^2*(n-1)!/(n-k)!^3 * b(k) * a(n-k)))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A007841, A249588, A269791, A269793, A269794.

Sequence in context: A012072 A007408 A066989 * A160501 A075987 A135099

Adjacent sequences: A249590 A249591 A249592 * A249594 A249595 A249596

Keyword

nonn

Author

Paul D. Hanna, Nov 02 2014

Extensions

Name clarified by Vaclav Kotesovec, Mar 05 2016

Status

approved