A035681 - OEIS

%I #11 Aug 15 2020 21:41:58

%S 0,0,0,0,1,1,1,1,2,2,2,2,5,5,5,5,8,8,8,8,14,15,15,15,22,23,23,23,34,

%T 37,38,38,51,54,55,55,74,81,84,85,108,116,119,120,151,165,172,175,214,

%U 230,238,241,291,317,332,339,402,434,451,459,538,584,613,628,727,784

%N Number of partitions of n into parts 8k+1 and 8k+4 with at least one part of each type.

%H Robert Price, <a href="/A035681/b035681.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f. : (-1 + 1/Product_{k>=0} (1 - x^(8 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 4). - _Robert Price_, Aug 15 2020

%t nmax = 66; s1 = Range[0, nmax/8]*8 + 1; s2 = Range[0, nmax/8]*8 + 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 = 66; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 15 2020*)

%Y Cf. A035441-A035468, A035618-A035680, A035682-A035699.

%K nonn

%O 1,9

%A _Olivier Gérard_