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A035665 Number of partitions of n into parts 7k+2 and 7k+6 with at least one part of each type.

%I A035665 #19 Aug 17 2020 07:15:05
%S A035665 0,0,0,0,0,0,0,1,0,1,0,1,0,2,2,2,2,2,2,3,4,6,4,7,4,8,6,12,10,13,12,14,
%T A035665 15,18,21,24,23,29,26,35,33,46,41,52,50,58,61,71,77,84,89,99,101,120,
%U A035665 121,146,142,168,166,192,199,226,239,259,275,300,316,354,370,416,422
%N A035665 Number of partitions of n into parts 7k+2 and 7k+6 with at least one part of each type.
%H A035665 Alois P. Heinz, <a href="/A035665/b035665.txt">Table of n, a(n) for n = 1..1000</a> (first 125 terms from Robert Price)
%F A035665 G.f.: (-1 + 1/Product_{k>=0} (1 - x^(7*k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(7*k + 6))). - _Robert Price_, Aug 16 2020
%p A035665 b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
%p A035665        b(n, i-1, t, s)+(h-> `if`(h in {2, 6}, add(b(n-i*j, i-1,
%p A035665       `if`(h=2, 1, t), `if`(h=6, 1, s)), j=1..n/i), 0))(irem(i, 7))))
%p A035665     end:
%p A035665 a:= n-> b(n$2, 0$2):
%p A035665 seq(a(n), n=1..75);  # _Alois P. Heinz_, Aug 14 2020
%t A035665 nmax = 71; s1 = Range[0, nmax/7]*7 + 2; s2 = Range[0, nmax/7]*7 + 6;
%t A035665 Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t A035665 x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 14 2020 *)
%t A035665 nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 16 2020 *)
%Y A035665 Cf. A035441-A035468, A035618-A035664, A035666-A035699.
%K A035665 nonn
%O A035665 1,14
%A A035665 _Olivier Gérard_