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%I #17 Aug 16 2020 17:16:35
%S 0,0,0,0,1,1,1,1,2,4,4,4,5,7,10,11,12,14,18,23,26,29,33,40,48,55,61,
%T 70,82,96,108,121,137,158,179,202,226,255,288,325,363,406,453,508,566,
%U 632,701,781,867,963,1066,1182,1306,1445,1592,1759,1939,2139,2350
%N Number of partitions of n into parts 5k+1 and 5k+4 with at least one part of each type.
%H Alois P. Heinz, <a href="/A035633/b035633.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Robert Price)
%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 4))). - _Robert Price_, Aug 16 2020
%t nmax = 59; s1 = Range[0, nmax/5]*5 + 1; s2 = Range[0, nmax/5]*5 + 4;
%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 = 59; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020 *)
%Y Cf. A035441-A035468, A035618-A035632, A035634-A035699.
%K nonn
%O 1,9
%A _Olivier Gérard_
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