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
Keywords
Examples
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)*...).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..180
Programs
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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 *) -
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), ", "))
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
Extensions
Name clarified by Vaclav Kotesovec, Mar 05 2016