A035669 - OEIS

%I #11 Aug 15 2020 21:40:51

%S 0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,2,1,1,1,2,3,1,4,3,3,3,5,7,4,7,7,8,8,10,

%T 13,10,14,15,16,17,20,25,21,26,29,32,33,37,45,41,47,54,58,61,65,79,76,

%U 83,94,103,108,113,132,135,143,160,172,185,192,219,227,240,265,286

%N Number of partitions of n into parts 7k+4 and 7k+5 with at least one part of each type.

%H Robert Price, <a href="/A035669/b035669.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 4)))*(-1 + 1/Product_{k>=0} (1 - x^(7 k + 5)). - _Robert Price_, Aug 15 2020

%t nmax = 74; s1 = Range[0, nmax/7]*7 + 4; s2 = Range[0, nmax/7]*7 + 5;

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

%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 15 2020 *)

%t nmax = 74; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 4)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 15 2020*)

%Y Cf. A035441-A035468, A035618-A035668, A035670-A035699.

%K nonn

%O 1,16

%A _Olivier Gérard_