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%I #17 Aug 16 2020 17:22:56
%S 0,0,0,0,1,0,1,1,1,3,2,3,4,4,7,6,9,10,11,16,15,20,23,25,32,34,41,47,
%T 52,63,68,80,90,101,116,129,147,166,184,210,232,262,292,326,363,405,
%U 450,501,554,617,681,756,834,924,1015,1125,1235,1363,1498,1647,1809
%N Number of partitions of n into parts 5k+2 and 5k+3 with at least one part of each type.
%H Alois P. Heinz, <a href="/A035634/b035634.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^(5 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 3))). - _Robert Price_, Aug 16 2020
%t nmax = 61; s1 = Range[0, nmax/5]*5 + 2; s2 = Range[0, nmax/5]*5 + 3;
%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 07 2020 *)
%t nmax = 61; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020 *)
%Y Cf. A035441-A035468, A035618-A035633, A035635-A035699.
%K nonn
%O 1,10
%A _Olivier Gérard_
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