A035676 - OEIS

%I #18 Aug 17 2020 07:10:15

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

%T 0,3,11,1,7,0,3,14,1,7,18,3,15,1,7,25,3,15,30,7,28,3,15,44,7,29,47,15,

%U 51,7,29,72,15,54,73,29,87,15,55,116,29,94,111,55,144,29,97,180,55

%N Number of partitions of n into parts 8k and 8k+5 with at least one part of each type.

%H Robert Price, <a href="/A035676/b035676.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 5)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - _Robert Price_, Aug 13 2020

%t nmax = 83; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 5;

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

%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 13 2020 *)

%t nmax = 83; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 13 2020 *)

%Y Cf. A035441-A035468, A035618-A035675, A035677-A035699.

%K nonn

%O 1,21

%A _Olivier Gérard_