A035689 - OEIS

%I #13 Aug 16 2020 14:23:04

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

%T 9,16,9,18,11,22,17,28,19,32,21,36,25,44,35,52,40,60,44,68,52,82,66,

%U 95,76,108,85,123,100,145,122,166,140,188,157,214,182,250,215,283,245

%N Number of partitions of n into parts 8k+2 and 8k+7 with at least one part of each type.

%H Alois P. Heinz, <a href="/A035689/b035689.txt">Table of n, a(n) for n = 1..5000</a>

%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 7))). - _Robert Price_, Aug 15 2020

%t nmax = 74; s1 = Range[0, nmax/8]*8 + 2; s2 = Range[0, nmax/8]*8 + 7;

%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 = 74; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 15 2020*)

%Y Cf. A035441-A035468, A035618-A035688, A035690-A035699.

%K nonn

%O 1,17

%A _Olivier Gérard_