A035627 Number of partitions of n into parts 5k and 5k+1 with at least one part of each type.

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 10, 11, 11, 11, 11, 22, 25, 26, 26, 26, 44, 51, 54, 55, 55, 84, 98, 105, 108, 109, 153, 178, 193, 200, 203, 270, 313, 341, 356, 363, 462, 532, 582, 611, 626, 771, 883, 968, 1021, 1050, 1259, 1431, 1571, 1663, 1717, 2017

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

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

Crossrefs

Cf. A035441-A035468, A035618-A035626, A035628-A035699.

Programs

Mathematica

nmax = 61; s1 = Range[1, nmax/5]*5; s2 = Range[0, nmax/5]*5 + 1;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 61; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 06 2020 *)

Formula

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