A035667 - OEIS

%I #20 Jan 03 2025 09:28:11

%S 0,0,0,0,0,0,0,1,0,0,1,0,1,1,2,1,1,3,1,3,3,4,4,3,7,4,7,8,8,10,8,14,11,

%T 14,18,16,20,19,27,24,28,35,31,40,40,48,48,53,64,60,73,74,86,90,96,

%U 114,108,129,135,149,159,167,196,190,221,234,249,274,285,326,324,367

%N Number of partitions of n into parts 7k+3 and 7k+5 with at least one part of each type.

%H Robert Price, <a href="/A035667/b035667.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(7*k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(7*k + 5))). - _Robert Price_, Aug 15 2020

%t nmax = 72; s1 = Range[0, nmax/7]*7 + 3; s2 = Range[0, nmax/7]*7 + 5;

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

%t nmax = 72; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 3)), {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-A035666, A035668-A035699.

%K nonn

%O 1,15

%A _Olivier Gérard_