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%I #11 Aug 15 2020 21:36:00
%S 0,0,0,0,0,0,0,0,1,0,0,1,0,0,2,2,0,2,2,0,3,4,3,3,4,4,4,6,8,8,6,9,11,8,
%T 13,18,13,14,21,19,18,29,30,25,32,38,35,40,51,52,50,59,66,67,73,89,93,
%U 89,104,119,115,129,156,154,153,186,195,195,228,254,251,269,304,319
%N Number of partitions of n into parts 7k+3 and 7k+6 with at least one part of each type.
%H Robert Price, <a href="/A035668/b035668.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f. : (-1 + 1/Product_{k>=0} (1 - x^(7 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(7 k + 6)). - _Robert Price_, Aug 15 2020
%t nmax = 74; s1 = Range[0, nmax/7]*7 + 3; s2 = Range[0, nmax/7]*7 + 6;
%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 = 74; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 15 2020*)
%Y Cf. A035441-A035468, A035618-A035667, A035669-A035699.
%K nonn
%O 1,15
%A _Olivier Gérard_
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