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%I #15 Aug 16 2020 17:15:24
%S 0,0,0,1,1,1,2,2,4,5,5,7,8,11,14,15,19,22,27,33,37,44,50,59,71,79,93,
%T 106,120,142,159,181,207,232,267,301,339,383,428,486,544,609,683,758,
%U 853,951,1056,1180,1304,1453,1616,1785,1980,2185,2417,2674,2947,3253
%N Number of partitions of n into parts 5k+1 and 5k+3 with at least one part of each type.
%H Robert Price, <a href="/A035632/b035632.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 3))). - _Robert Price_, Aug 16 2020
%t nmax = 58; s1 = Range[0, nmax/5]*5 + 1; s2 = Range[0, nmax/5]*5 + 3;
%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 07 2020 *)
%t nmax = 58; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020 *)
%Y Cf. A035441-A035468, A035618-A035631, A035633-A035699.
%K nonn
%O 1,7
%A _Olivier Gérard_
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