cp's OEIS Frontend

A035664 Number of partitions of n into parts 7k+2 and 7k+5 with at least one part of each type.

%I A035664 #16 Aug 16 2020 21:19:11
%S A035664 0,0,0,0,0,0,1,0,1,0,1,1,1,3,1,3,2,3,4,3,7,4,8,6,8,10,9,15,11,17,15,
%T A035664 19,22,21,29,25,35,33,39,44,44,57,52,67,65,76,84,86,103,101,122,124,
%U A035664 139,153,158,185,185,216,222,247,268,282,317,327,369,387,422,458,482
%N A035664 Number of partitions of n into parts 7k+2 and 7k+5 with at least one part of each type.
%H A035664 Alois P. Heinz, <a href="/A035664/b035664.txt">Table of n, a(n) for n = 1..1000</a> (first 125 terms from Robert Price)
%F A035664 G.f.: (-1 + 1/Product_{k>=0} (1 - x^(7 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(7 k + 5))). - _Robert Price_, Aug 16 2020
%p A035664 b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
%p A035664        b(n, i-1, t, s)+(h-> `if`(h in {2, 5}, add(b(n-i*j, i-1,
%p A035664       `if`(h=2, 1, t), `if`(h=5, 1, s)), j=1..n/i), 0))(irem(i, 7))))
%p A035664     end:
%p A035664 a:= n-> b(n$2, 0$2):
%p A035664 seq(a(n), n=1..75);  # _Alois P. Heinz_, Aug 14 2020
%t A035664 nmax = 69; s1 = Range[0, nmax/7]*7 + 2; s2 = Range[0, nmax/7]*7 + 5;
%t A035664 Table[Count[IntegerPartitions[n, All, s1~Join~s2],
%t A035664 x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 14 2020 *)
%t A035664 nmax = 69; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 16 2020 *)
%Y A035664 Cf. A035441-A035468, A035618-A035663, A035665-A035699.
%K A035664 nonn
%O A035664 1,14
%A A035664 _Olivier Gérard_