%I #17 Aug 16 2020 20:14:24

%S 0,0,0,0,0,0,1,0,1,0,1,1,3,1,3,1,4,3,7,3,8,4,10,8,14,9,17,11,22,17,28,

%T 20,34,25,43,35,53,42,64,51,80,67,96,80,115,98,142,123,168,147,200,

%U 178,244,217,286,257,339,310,407,371,475,439,559,523,664,618,772,726

%N Number of partitions of n into parts 6k+2 and 6k+5 with at least one part of each type.

%H Alois P. Heinz, <a href="/A035648/b035648.txt">Table of n, a(n) for n = 1..5000</a> (first 100 terms from Robert Price)

%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 5))). - _Robert Price_, Aug 16 2020

%t nmax = 68; s1 = Range[0, nmax/6]*6 + 2; s2 = Range[0, nmax/6]*6 + 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 = 68; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020 *)

%Y Cf. A035441-A035468, A035618-A035647, A035649-A035699.

%K nonn

%O 1,13

%A _Olivier Gérard_