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%I A035650 #16 Aug 16 2020 20:15:23
%S A035650 0,0,0,0,0,0,0,1,0,0,1,0,1,3,0,1,3,1,3,6,1,3,8,3,7,12,3,9,16,7,15,22,
%T A035650 9,19,30,16,29,40,20,38,54,32,54,69,41,70,93,61,95,118,78,124,156,110,
%U A035650 163,195,141,211,255,192,271,317,245,349,409,323,441,506,412,562,644
%N A035650 Number of partitions of n into parts 6k+3 and 6k+5 with at least one part of each type.
%H A035650 Alois P. Heinz, <a href="/A035650/b035650.txt">Table of n, a(n) for n = 1..1000</a> (first 125 terms from Robert Price)
%F A035650 G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 6))). - _Robert Price_, Aug 16 2020
%p A035650 b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
%p A035650        b(n, i-1, t, s)+(h-> `if`(h in {3, 5}, add(b(n-i*j, i-1,
%p A035650       `if`(h=3, 1, t), `if`(h=5, 1, s)), j=1..n/i), 0))(irem(i, 6))))
%p A035650     end:
%p A035650 a:= n-> b(n$2, 0$2):
%p A035650 seq(a(n), n=1..75);  # _Alois P. Heinz_, Aug 14 2020
%t A035650 nmax = 71; s1 = Range[0, nmax/6]*6 + 3; s2 = Range[0, nmax/6]*6 + 5;
%t A035650 Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t A035650 x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 14 2020 *)
%t A035650 nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 16 2020 *)
%Y A035650 Cf. A035441-A035468, A035618-A035649, A035651-A035699.
%K A035650 nonn
%O A035650 1,14
%A A035650 _Olivier Gérard_

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