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%I #12 Aug 16 2020 14:08:42
%S 0,0,0,0,0,0,1,0,0,1,1,0,1,1,3,1,1,3,4,1,3,4,7,3,4,8,10,4,8,11,15,8,
%T 11,18,21,11,19,24,30,19,25,37,42,25,40,50,56,41,53,70,79,54,77,95,
%U 103,80,103,129,141,106,144,172,183,151,189,228,246,197,257,301,314
%N Number of partitions of n into parts 8k+3 and 8k+4 with at least one part of each type.
%H Alois P. Heinz, <a href="/A035690/b035690.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 + 4))). - _Robert Price_, Aug 15 2020
%t nmax = 71; s1 = Range[0, nmax/8]*8 + 3; s2 = Range[0, nmax/8]*8 + 4;
%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 = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 15 2020*)
%Y Cf. A035441-A035468, A035618-A035689, A035691-A035699.
%K nonn
%O 1,15
%A _Olivier Gérard_
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