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%I #14 Aug 15 2020 21:43:27
%S 0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,4,4,4,4,4,4,5,7,10,11,11,11,11,12,14,
%T 18,23,25,26,26,27,29,33,40,47,52,54,56,58,62,70,81,93,101,107,111,
%U 116,124,137,155,172,188,199,208,218,233,255,282,311,336,357,374,393
%N Number of partitions of n into parts 8k+1 and 8k+7 with at least one part of each type.
%H Alois P. Heinz, <a href="/A035684/b035684.txt">Table of n, a(n) for n = 1..5000</a>
%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 7). - _Robert Price_, Aug 15 2020
%t nmax = 68; s1 = Range[0, nmax/8]*8 + 1; s2 = Range[0, nmax/8]*8 + 7;
%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 = 68; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 7)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 15 2020*)
%Y Cf. A035441-A035468, A035618-A035683, A035685-A035699.
%K nonn
%O 1,15
%A _Olivier Gérard_
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