A363239 -id:A363239 - cp's OEIS frontend

A363233 Number of partitions of n with rank a multiple of 4.

1, 0, 1, 1, 3, 1, 5, 4, 10, 8, 16, 17, 29, 29, 48, 53, 81, 89, 130, 149, 208, 238, 325, 381, 506, 592, 770, 910, 1165, 1374, 1738, 2057, 2571, 3038, 3761, 4451, 5461, 6447, 7855, 9270, 11219, 13214, 15899, 18703, 22386, 26276, 31306, 36691, 43525, 50902, 60149, 70221, 82679, 96325

Offset: 1

Seiichi Manyama, May 23 2023

nonn

Cf. A000041, A328988, A340601, A363237, A363238, A363239.

b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 4)=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..54);  # Alois P. Heinz, May 23 2023

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^(4*k))/(1-x^(4*k))))

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^(4*k)) / (1-x^(4*k)).

A363237 Number of partitions of n with rank a multiple of 5.

Original entry on oeis.org

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

Seiichi Manyama, May 23 2023

nonn

Cf. A000041, A328988, A340601, A363233, A363238, A363239.

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

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))))

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)).

A363238 Number of partitions of n with rank a multiple of 6.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 5, 2, 6, 6, 10, 11, 21, 19, 32, 37, 51, 59, 90, 97, 138, 162, 215, 253, 340, 392, 514, 610, 771, 916, 1166, 1367, 1711, 2032, 2503, 2965, 3647, 4293, 5237, 6188, 7469, 8808, 10613, 12459, 14920, 17530, 20862, 24457, 29029, 33924, 40099, 46829, 55101, 64215, 75386

Offset: 1

Seiichi Manyama, May 23 2023

nonn

Cf. A000041, A328988, A340601, A363233, A363237, A363239.

b:= proc(n, i, c) option remember; `if`(i>n, 0, `if`(i=n,
     `if`(irem(i-c, 6)=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

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^(6*k))/(1-x^(6*k))))

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^(6*k)) / (1-x^(6*k)).

cp's OEIS Frontend

A363233 Number of partitions of n with rank a multiple of 4.

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A363237 Number of partitions of n with rank a multiple of 5.

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A363238 Number of partitions of n with rank a multiple of 6.

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