A035666 - OEIS

%I #17 Jan 03 2025 10:57:23

%S 0,0,0,0,0,0,1,0,0,1,1,0,1,3,1,1,3,3,2,3,6,4,4,7,8,6,8,13,10,10,16,17,

%T 14,19,25,22,23,32,34,31,38,48,45,47,60,65,62,73,86,86,90,109,117,117,

%U 133,153,155,165,191,205,209,235,261,272,288,326,349,362,398,440,459

%N Number of partitions of n into parts 7k+3 and 7k+4 with at least one part of each type.

%H Alois P. Heinz, <a href="/A035666/b035666.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 + 4))). - _Robert Price_, Aug 15 2020

%t nmax = 71; s1 = Range[0, nmax/7]*7 + 3; 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 15 2020 *)

%t nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 15 2020*)

%Y Cf. A035441-A035468, A035618-A035665, A035667-A035699.

%K nonn

%O 1,14

%A _Olivier Gérard_