A035630 - OEIS

%I #19 Aug 16 2020 17:07:42

%S 0,0,0,0,0,0,0,0,1,0,0,0,1,3,0,0,1,3,6,0,1,3,7,11,1,3,7,14,19,3,7,15,

%T 26,32,7,15,29,46,51,15,30,53,76,81,30,56,91,124,126,57,98,152,195,

%U 195,101,167,245,304,296,174,274,388,461,448,289,441,598,696,668,470

%N Number of partitions of n into parts 5k and 5k+4 with at least one part of each type.

%H Robert Price, <a href="/A035630/b035630.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: (-1 + 1/Product_{k>=0} (1-x^(5k+4)))*(-1 + 1/Product_{k>=1} (1-x^(5k))). - _Robert Price_, Aug 07 2020

%t nmax = 70; s1 = Range[1, nmax/5]*5; s2 = Range[0, nmax/5]*5 + 4;

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

%t x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* _Robert Price_, Aug 06 2020 *)

%t nmax = 70; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* _Robert Price_, Aug 06 2020 *)

%Y Cf. A035441-A035468, A035618-A035629, A035631-A035699.

%K nonn

%O 1,14

%A _Olivier Gérard_