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%I #16 Aug 16 2020 20:14:52
%S 0,0,0,0,0,0,1,0,0,1,1,0,3,1,1,3,3,1,7,3,3,8,8,3,14,9,8,16,17,9,27,19,
%T 18,32,34,20,49,40,37,58,63,43,87,74,70,104,113,82,149,135,128,177,
%U 195,152,249,232,224,298,327,266,407,392,380,485,535,455,654,639,628
%N Number of partitions of n into parts 6k+3 and 6k+4 with at least one part of each type.
%H Alois P. Heinz, <a href="/A035649/b035649.txt">Table of n, a(n) for n = 1..1000</a> (first 125 terms from Robert Price)
%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 4))). - _Robert Price_, Aug 16 2020
%p b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
%p b(n, i-1, t, s)+(h-> `if`(h in {3, 4}, add(b(n-i*j, i-1,
%p `if`(h=3, 1, t), `if`(h=4, 1, s)), j=1..n/i), 0))(irem(i, 6))))
%p end:
%p a:= n-> b(n$2, 0$2):
%p seq(a(n), n=1..75); # _Alois P. Heinz_, Aug 14 2020
%t nmax = 69; s1 = Range[0, nmax/6]*6 + 3; s2 = Range[0, nmax/6]*6 + 4;
%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 14 2020 *)
%t nmax = 69; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020 *)
%Y Cf. A035441-A035468, A035618-A035648, A035650-A035699.
%K nonn
%O 1,13
%A _Olivier Gérard_
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