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%I #15 Aug 16 2020 16:31:30
%S 0,0,0,1,1,1,4,4,4,10,11,11,22,25,26,44,51,54,84,98,105,152,178,193,
%T 266,312,341,452,528,581,749,873,964,1214,1409,1561,1930,2234,2479,
%U 3018,3478,3866,4647,5339,5937,7061,8081,8991,10594,12089,13447,15721
%N Number of partitions of n into parts 3k and 3k+1 with at least one part of each type.
%H Alois P. Heinz, <a href="/A035618/b035618.txt">Table of n, a(n) for n = 1..1000</a> (first 75 terms from Robert Price)
%F G.f.: (-1 + 1/Product_{k>=1} (1 - x^(3 k)))*(-1 + 1/Product_{k>=0} (1 - x^(3 k + 1))). - _Robert Price_, Aug 16 2020
%t nmax = 52; kmax = nmax/3; s1 = Range[1, nmax/3]*3; s2 = Range[0, nmax/3]*3 + 1;
%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 06 2020 *)
%t nmax = 52; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(3 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(3 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020*)
%Y Cf. A035441-A035468, A035619-A035699.
%K nonn
%O 1,7
%A _Olivier Gérard_
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