cp's OEIS Frontend

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

%I A035678 #21 Aug 17 2020 06:48:18
%S A035678 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,
%T A035678 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,
%U A035678 1,3,7,15,29,50,69,67,3,7,15,29,53,84,110,99,7,15,29,54,91,138,168
%N A035678 Number of partitions of n into parts 8k and 8k+7 with at least one part of each type.
%H A035678 Robert Israel, <a href="/A035678/b035678.txt">Table of n, a(n) for n = 1..5000</a>
%F A035678 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
%p A035678 np:= combinat:-numbpart:
%p A035678 NP:= proc(n,m) if m > n then np(n) else np(n,m) fi end proc;
%p A035678 f:= proc(n) local r0;
%p A035678    r0:= (-n) mod 8;
%p A035678    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))
%p A035678 end proc:
%p A035678 seq(f(n),n=1..100); # _Robert Israel_, Apr 06 2016
%t A035678 nmax = 86; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 7;
%t A035678 Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t A035678 x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 13 2020 *)
%t A035678 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 *)
%Y A035678 Cf. A035441-A035468, A035618-A035677, A035679-A035699.
%K A035678 nonn
%O A035678 1,23
%A A035678 _Olivier Gérard_