A035656 - OEIS

%I #14 Aug 16 2020 20:27:47

%S 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,3,0,0,0,0,1,3,6,0,0,0,1,3,7,11,

%T 0,0,1,3,7,14,18,0,1,3,7,15,25,29,1,3,7,15,28,43,45,3,7,15,29,50,70,

%U 69,7,15,29,53,85,112,103,15,29,54,92,140,172,153,29,54,95,155,222

%N Number of partitions of n into parts 7k and 7k+6 with at least one part of each type.

%H Robert Price, <a href="/A035656/b035656.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f. : (-1 + 1/Product_ {k >= 0} (1 - x^(7 k + 6)))*(-1 + 1/Product_ {k >= 1} (1 - x^(7 k))). - _Robert Price_, Aug 12 2020

%t nmax = 81; s1 = Range[1, nmax/7]*7; s2 = Range[0, nmax/7]*7 + 6;

%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],

%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 12 2020 *)

%t nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 12 2020 *)

%Y Cf. A035441-A035468, A035618-A035655, A035657-A035699.

%K nonn

%O 1,20

%A _Olivier Gérard_