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

%I #14 Aug 16 2020 17:06:50

%S 0,0,0,0,0,1,1,1,1,1,4,4,4,4,4,10,11,11,11,11,22,25,26,26,26,44,51,54,

%T 55,55,84,98,105,108,109,153,178,193,200,203,270,313,341,356,363,462,

%U 532,582,611,626,771,883,968,1021,1050,1259,1431,1571,1663,1717,2017

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

%H Robert Price, <a href="/A035627/b035627.txt">Table of n, a(n) for n = 1..1000</a>

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

%t nmax = 61; s1 = Range[1, nmax/5]*5; s2 = Range[0, nmax/5]*5 + 1;

%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 = 61; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 06 2020 *)

%Y Cf. A035441-A035468, A035618-A035626, A035628-A035699.

%K nonn

%O 1,11

%A _Olivier GĂ©rard_