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A363239
Number of partitions of n with rank a multiple of 7.
3
1, 0, 1, 1, 1, 1, 3, 4, 4, 6, 8, 11, 15, 19, 26, 33, 43, 55, 70, 89, 114, 144, 179, 225, 280, 348, 430, 532, 653, 800, 978, 1193, 1449, 1758, 2127, 2569, 3091, 3717, 4455, 5334, 6369, 7596, 9039, 10739, 12734, 15080, 17822, 21039, 24791, 29176, 34277, 40227, 47133, 55165, 64468
OFFSET
1,7
FORMULA
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2) * (1-x^k) * (1+x^(7*k)) / (1-x^(7*k)).
MAPLE
b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
`if`(irem(i-c, 7)=0, 1, 0), b(n-i, i, c+1)+b(n, i+1, c)))
end:
a:= n-> b(n, 1$2):
seq(a(n), n=1..55); # Alois P. Heinz, May 23 2023
PROG
(PARI) my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1-x^k)*(1+x^(7*k))/(1-x^(7*k))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 23 2023
STATUS
approved