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%I #14 Aug 16 2020 12:59:50
%S 0,0,0,0,0,0,0,1,0,0,1,0,1,1,0,3,1,1,3,1,3,3,2,6,3,4,7,4,8,7,6,13,8,
%T 10,15,10,17,17,14,24,19,22,30,23,33,34,31,46,39,44,56,47,63,65,61,82,
%U 75,84,101,90,113,118,115,145,137,151,176,165,197,207,206,246,242,264
%N Number of partitions of n into parts 8k+3 and 8k+5 with at least one part of each type.
%H Alois P. Heinz, <a href="/A035691/b035691.txt">Table of n, a(n) for n = 1..5000</a>
%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 5))). - _Robert Price_, Aug 15 2020
%t nmax = 74; s1 = Range[0, nmax/8]*8 + 3; 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 15 2020 *)
%t nmax = 74; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 15 2020*)
%Y Cf. A035441-A035468, A035618-A035690, A035692-A035699.
%K nonn
%O 1,16
%A _Olivier Gérard_
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