OFFSET
0,6
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..100 from Antonio Roldán)
FORMULA
From Vaclav Kotesovec, Mar 11 2026: (Start)
G.f.: Product_{k>=1} 1/(1 - x^(k*(3*k-1)/2)).
a(n) ~ Gamma(1 + b/d) * zeta(3/2)^(2/3 + b/(3*d)) * d^(1/6 + b/(3*d)) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / (2^(4/3)*d^(1/3))) / (sqrt(3)*2^(7/3 + 2*b/(3*d)) * Pi^(7/6 - b/(6*d)) * n^(7/6 + b/(3*d))) * (1 - (136*d^2 + 120*d*b + 3*b^2*(8 - 3*Pi*zeta(1/2)*zeta(3/2))) / (72 * d^(5/3) * Pi^(1/3) * (2*zeta(3/2))^(2/3) * n^(1/3))), where d = 3/2, b = -1/2. (End)
EXAMPLE
A(15)=5 because 15 = 12+1+1+1 = 5+5+5 = 5+5+1+1+1+1+1 = 5+1+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1 with 12, 5, 1 pentagonal numbers.
MATHEMATICA
nmax = 100; CoefficientList[Series[1/Product[1 - x^(k*(3*k-1)/2), {k, 1, Floor[Sqrt[1 + 24*nmax]/6 + 1]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2026 *)
PROG
(PARI) {for (n=1, 100, p=truncate((1+sqrt(24*n+1))/6); m=polcoeff(prod(k=1, p, q=(3*k-1)*k/2; sum(h=0, truncate(n/q+1), x^(h*q))), n); write("B218379.txt", n, " ", m))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Antonio Roldán, Oct 27 2012
EXTENSIONS
a(0) = 1 prepended by Seiichi Manyama, Dec 09 2017
STATUS
approved
