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Number of partitions of n into parts 5k+3 and 5k+4 with at least one part of each type.
3

%I #15 Aug 16 2020 18:36:34

%S 0,0,0,0,0,0,1,0,0,1,1,2,1,1,3,3,4,3,4,7,7,8,8,10,14,14,16,18,20,27,

%T 28,30,35,40,48,52,55,64,73,85,90,98,114,128,143,155,168,195,214,237,

%U 259,283,319,353,385,422,460,516,564,618,672,734,816,892,964,1057

%N Number of partitions of n into parts 5k+3 and 5k+4 with at least one part of each type.

%H Robert Price, <a href="/A035636/b035636.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 4))). - _Robert Price_, Aug 16 2020

%t nmax = 66; s1 = Range[0, nmax/5]*5 + 3; 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 07 2020 *)

%t nmax = 66; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020 *)

%Y Cf. A035441-A035468, A035618-A035635, A035637-A035699.

%K nonn

%O 1,12

%A _Olivier GĂ©rard_