A035638 -id:A035638 - cp's OEIS frontend

A035637 Number of partitions of n into parts 6k and 6k+1 with at least one part of each type.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 10, 11, 11, 11, 11, 11, 22, 25, 26, 26, 26, 26, 44, 51, 54, 55, 55, 55, 84, 98, 105, 108, 109, 109, 153, 178, 193, 200, 203, 204, 270, 313, 341, 356, 363, 366, 463, 532, 582, 611, 626, 633, 774, 884, 968, 1021

Offset: 1

Olivier Gérard

nonn

Robert Price, Table of n, a(n) for n = 1..1000

Cf. A035441-A035468, A035618-A035636, A035638-A035699.

nmax = 64; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 1;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 07 2020 *)
nmax = 64; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 07 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1-x^(6k+1)))*(-1 + 1/Product_{k>=1} (1-x^(6k))). - Robert Price, Aug 07 2020

A035639 Number of partitions of n into parts 6k and 6k+3 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 4, 0, 0, 4, 0, 0, 10, 0, 0, 11, 0, 0, 22, 0, 0, 25, 0, 0, 44, 0, 0, 51, 0, 0, 83, 0, 0, 98, 0, 0, 149, 0, 0, 177, 0, 0, 259, 0, 0, 309, 0, 0, 436, 0, 0, 521, 0, 0, 716, 0, 0, 857, 0, 0, 1151, 0, 0, 1376, 0, 0, 1816, 0, 0, 2170, 0, 0, 2818, 0, 0

Offset: 1

Olivier Gérard

nonn

Robert Price, Table of n, a(n) for n = 1..999

Cf. A035441-A035468, A035618-A035638, A035640-A035699.

First trisection gives A006477.

nmax = 83; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(6 k + 3)))*(-1 + 1/Product_{k>=1} (1 - x^(6 k))). - Robert Price, Aug 12 2020

cp's OEIS Frontend

A035637 Number of partitions of n into parts 6k and 6k+1 with at least one part of each type.

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