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A035623
Number of partitions of n into parts 4k and 4k+3 with at least one part of each type.
4
0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 1, 3, 6, 1, 3, 7, 12, 3, 7, 15, 21, 7, 16, 28, 36, 16, 31, 50, 60, 32, 57, 85, 98, 60, 100, 141, 157, 107, 169, 226, 248, 184, 276, 358, 385, 305, 442, 553, 591, 495, 691, 845, 896, 782, 1063, 1270, 1343, 1216, 1608, 1890, 1993
OFFSET
1,11
LINKS
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1-x^(4k+3)))*(-1 + 1/Product_{k>=1} (1-x^(4k))). - Robert Israel, Feb 23 2016
a(n) ~ exp(Pi*sqrt(n/3)) * Pi^(3/4) / (2^(5/4) * 3^(5/8) * Gamma(1/4) * n^(9/8)). - Vaclav Kotesovec, May 26 2018
MAPLE
N:= 100:
P:= (-1 + 1/mul(1-x^(4*k+3), k=0..(N-3)/4))*(-1 + 1/mul(1-x^(4*k), k=1..N/4)):
S:= series(P, x, N+1):
seq(coeff(S, x, j), j=1..N); # Robert Israel, Feb 23 2016
MATHEMATICA
nmax = 63; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 63; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 06 2020 *)
KEYWORD
nonn
STATUS
approved