login
A363237
Number of partitions of n with rank a multiple of 5.
3
1, 0, 1, 1, 1, 3, 3, 4, 6, 8, 12, 15, 21, 27, 34, 47, 59, 77, 98, 125, 160, 200, 251, 315, 390, 488, 602, 744, 913, 1120, 1370, 1669, 2029, 2462, 2975, 3597, 4327, 5203, 6237, 7466, 8919, 10634, 12653, 15035, 17824, 21114, 24950, 29455, 34705, 40844, 47991, 56317, 65987, 77231, 90252
OFFSET
1,6
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^(5*k)) / (1-x^(5*k)).
MAPLE
b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
`if`(irem(i-c, 5)=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^(5*k))/(1-x^(5*k))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 23 2023
STATUS
approved