A035633 Number of partitions of n into parts 5k+1 and 5k+4 with at least one part of each type.

0, 0, 0, 0, 1, 1, 1, 1, 2, 4, 4, 4, 5, 7, 10, 11, 12, 14, 18, 23, 26, 29, 33, 40, 48, 55, 61, 70, 82, 96, 108, 121, 137, 158, 179, 202, 226, 255, 288, 325, 363, 406, 453, 508, 566, 632, 701, 781, 867, 963, 1066, 1182, 1306, 1445, 1592, 1759, 1939, 2139, 2350

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert Price)

Crossrefs

Cf. A035441-A035468, A035618-A035632, A035634-A035699.

Programs

Mathematica

nmax = 59; s1 = Range[0, nmax/5]*5 + 1; s2 = Range[0, nmax/5]*5 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 07 2020 *)
nmax = 59; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 4))). - Robert Price, Aug 16 2020