A035654 - OEIS

%I #16 Aug 16 2020 20:27:12

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

%T 1,14,7,3,19,15,7,3,26,15,7,32,29,15,7,46,30,15,51,53,30,15,76,56,30,

%U 81,91,57,30,124,98,57,126,152,101,57,195,167,102,195,245,174,102,304

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

%H Robert Price, <a href="/A035654/b035654.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>=1} (1 - x^(7 k))). - _Robert Price_, Aug 12 2020

%t nmax = 78; s1 = Range[1, nmax/7]*7; s2 = Range[0, nmax/7]*7 + 4;

%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 = 78; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 12 2020 *)

%Y Cf. A035441-A035468, A035618-A035653, A035655-A035699.

%K nonn

%O 1,18

%A _Olivier Gérard_