OFFSET
0,36
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 + x^(k*(3*k-1)/2)).
a(n) ~ zeta(3/2)^(1/3) * (sqrt(2) - 1)^(1/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * (sqrt(2) - 1)^(2/3) * n^(1/3) / (2^(5/3)*d^(1/3))) / (2^(4/3 + b/(2*d)) * sqrt(3) * d^(1/6) * Pi^(1/3) * n^(5/6)) * (1 - ((sqrt(2) - 1)^(4/3) * b^2 * Pi^(2/3) * zeta(1/2) * zeta(3/2)^(1/3) / (2^(23/6) * d^(5/3)) + 5*d^(1/3) / (9 * (2*Pi)^(1/3) * (sqrt(2) - 1)^(2/3) * zeta(3/2)^(2/3))) / n^(1/3)), where d = 3/2, b = -1/2. (End)
EXAMPLE
A(98)=3 because 98 = 12 + 35 + 51 = 1 + 5 + 92 = 1 + 5 + 22 + 70 with 1, 5, 22, 70, 92 pentagonal numbers.
MATHEMATICA
nmax = 100; CoefficientList[Series[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, m=polcoeff(prod(k=1, truncate(1+sqrt(24*n+1))/6, 1+x^(k*(3*k-1)/2)), n); write("B218380.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
