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Number of partitions of n into parts 4k and 4k+1 with at least one part of each type.
4

%I #15 Aug 16 2020 16:32:14

%S 0,0,0,0,1,1,1,1,4,4,4,4,10,11,11,11,22,25,26,26,44,51,54,55,84,98,

%T 105,108,153,178,193,200,269,313,341,356,459,531,582,611,764,880,967,

%U 1021,1244,1424,1568,1662,1988,2264,2494,2653,3122,3536,3896,4155

%N Number of partitions of n into parts 4k and 4k+1 with at least one part of each type.

%H Alois P. Heinz, <a href="/A035621/b035621.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Robert Price)

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

%t nmax = 56; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 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 = 56; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x] (* _Robert Price_, Aug 16 2020*)

%Y Cf. A035441-A035468, A035618-A035620, A035622-A035699.

%K nonn

%O 1,9

%A _Olivier GĂ©rard_