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%I #11 Aug 15 2020 21:41:34
%S 0,0,0,1,1,1,2,2,2,3,3,5,6,6,8,9,9,11,12,15,18,19,23,26,27,31,34,39,
%T 45,49,56,62,66,74,80,89,101,109,123,136,144,160,173,187,210,227,249,
%U 275,293,319,346,371,408,442,480,525,562,608,655,701,763,822,887,963
%N Number of partitions of n into parts 8k+1 and 8k+3 with at least one part of each type.
%H Robert Price, <a href="/A035680/b035680.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f. : (-1 + 1/Product_{k>=0} (1 - x^(8 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(8 k + 3). - _Robert Price_, Aug 15 2020
%t nmax = 64; s1 = Range[0, nmax/8]*8 + 1; s2 = Range[0, nmax/8]*8 + 3;
%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 = 64; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 15 2020*)
%Y Cf. A035441-A035468, A035618-A035679, A035681-A035699.
%K nonn
%O 1,7
%A _Olivier Gérard_
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