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%I #16 Aug 16 2020 20:10:16
%S 0,0,0,0,0,1,1,1,1,1,2,4,4,4,4,5,7,10,11,11,12,14,18,23,25,27,29,33,
%T 40,47,53,57,62,70,81,94,104,113,123,137,156,175,194,211,230,255,285,
%U 317,348,379,413,454,502,552,604,657,715,782,857,937,1021,1109,1203
%N Number of partitions of n into parts 6k+1 and 6k+5 with at least one part of each type.
%H Alois P. Heinz, <a href="/A035645/b035645.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 + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 5))). - _Robert Price_, Aug 16 2020
%t nmax = 63; s1 = Range[0, nmax/6]*6 + 1; 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 = 63; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 1)), {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-A035644, A035646-A035699.
%K nonn
%O 1,11
%A _Olivier Gérard_
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