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Number of partitions of n into parts 3k and 3k+2 with at least one part of each type.
4

%I #15 Aug 16 2020 16:31:45

%S 0,0,0,0,1,0,1,3,1,3,7,3,8,14,8,17,26,18,33,47,36,61,81,68,106,137,

%T 121,181,224,209,296,362,347,478,570,565,750,890,894,1166,1360,1396,

%U 1774,2062,2134,2677,3076,3228,3973,4555,4804,5854,6657,7085,8513

%N Number of partitions of n into parts 3k and 3k+2 with at least one part of each type.

%H Alois P. Heinz, <a href="/A035619/b035619.txt">Table of n, a(n) for n = 1..1000</a> (first 80 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 + 2))). - _Robert Price_, Aug 16 2020

%t nmax = 55; s1 = Range[1, nmax/3]*3; s2 = Range[0, nmax/3]*3 + 2;

%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 = 55; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(3 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(3 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020*)

%Y Cf. A035441-A035468, A035618, A035620-A035699.

%K nonn

%O 1,8

%A _Olivier GĂ©rard_