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Number of partitions of n into parts 5k+1 and 5k+2 with at least one part of each type.
3

%I #17 Aug 16 2020 17:13:33

%S 0,0,1,1,2,2,3,5,6,8,9,11,15,18,23,26,31,38,45,55,62,73,85,100,118,

%T 135,155,178,205,238,269,308,347,397,452,512,578,650,733,829,929,1045,

%U 1164,1307,1461,1635,1822,2026,2254,2511,2787,3098,3422,3796,4196

%N Number of partitions of n into parts 5k+1 and 5k+2 with at least one part of each type.

%H Robert Price, <a href="/A035631/b035631.txt">Table of n, a(n) for n = 1..1000</a>

%F G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 2))). - _Robert Price_, Aug 16 2020

%t nmax = 55; s1 = Range[0, nmax/5]*5 + 1; s2 = Range[0, nmax/5]*5 + 2;

%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 = 55; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020 *)

%Y Cf. A035371.

%Y Cf. A035441-A035468, A035618-A035630, A035632-A035699.

%K nonn

%O 1,5

%A _Olivier GĂ©rard_