A035678 Number of partitions of n into parts 8k and 8k+7 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 3, 6, 0, 0, 0, 0, 1, 3, 7, 11, 0, 0, 0, 1, 3, 7, 14, 18, 0, 0, 1, 3, 7, 15, 25, 29, 0, 1, 3, 7, 15, 28, 43, 44, 1, 3, 7, 15, 29, 50, 69, 67, 3, 7, 15, 29, 53, 84, 110, 99, 7, 15, 29, 54, 91, 138, 168

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A035441-A035468, A035618-A035677, A035679-A035699.

Programs

Maple

np:= combinat:-numbpart:
NP:= proc(n,m) if m > n then np(n) else np(n,m) fi end proc;
f:= proc(n) local r0;
   r0:= (-n) mod 8;
   add(np(s)*add(NP((n-8*s-7*r)/8, r), r=r0 .. floor((n-8*s)/7), 8), s=1..floor((n-1)/8))
end proc:
seq(f(n),n=1..100); # Robert Israel, Apr 06 2016

Mathematica

nmax = 86; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 7;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 86; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 13 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 7)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - Robert Price, Aug 13 2020