A035673 - OEIS

%I #18 Aug 17 2020 07:20:46

%S 0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,4,0,4,0,4,0,4,0,10,0,11,0,11,0,11,

%T 0,22,0,25,0,26,0,26,0,44,0,51,0,54,0,55,0,84,0,98,0,105,0,108,0,153,

%U 0,178,0,193,0,200,0,269,0,313,0,341,0,356,0,459,0,531,0,582,0,611,0

%N Number of partitions of n into parts 8k and 8k+2 with at least one part of each type.

%H Robert Price, <a href="/A035673/b035673.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(8*k + 2)))*(-1 + 1/Product_{k>=1} (1 - x^(8*k))). - _Robert Price_, Aug 12 2020

%t nmax = 81; s1 = Range[1, nmax/8]*8; s2 = Range[0, nmax/8]*8 + 2;

%t Table[Count[IntegerPartitions[n, All, s1~Join~s2],

%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 12 2020 *)

%t nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(8 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(8 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 12 2020 *)

%Y Cf. A035441-A035468, A035618-A035672, A035674-A035699.

%Y Bisections give: A035621 (even part), A000004 (odd part).

%K nonn

%O 1,18

%A _Olivier Gérard_