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A318027
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Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(4*k))).
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2
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1, 1, 2, 3, 6, 8, 13, 18, 29, 39, 57, 77, 112, 148, 205, 271, 372, 484, 647, 838, 1110, 1423, 1852, 2361, 3051, 3857, 4922, 6191, 7849, 9805, 12319, 15314, 19131, 23649, 29333, 36099, 44556, 54568, 66963, 81683, 99803, 121229, 147413, 178411, 216111, 260590, 314365, 377819, 454229
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OFFSET
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0,3
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COMMENTS
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Convolution of A000041 and A035444.
Convolution of A000712 and A082303.
Convolution inverse of A107034.
Number of partitions of n if there are 2 kinds of parts that are multiples of 4.
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LINKS
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Table of n, a(n) for n=0..48.
Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
Index entries for sequences related to partitions
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FORMULA
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G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + 2*x^(3*k))/(k*(1 - x^(4*k)))).
a(n) ~ 5^(3/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018
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EXAMPLE
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a(5) = 8 because we have [5], [4, 1], [4', 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].
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MAPLE
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a:=series(mul(1/((1-x^k)*(1-x^(4*k))), k=1..55), x=0, 49): seq(coeff(a, x, n), n=0..48); # Paolo P. Lava, Apr 02 2019
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MATHEMATICA
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nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(4 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^4]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + 2*x^(3 k))/(k (1 - x^(4 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 4 k], {k, 0, n/4}], {n, 0, 48}]
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CROSSREFS
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Cf. A000041, A000712, A002512 (self-convolution), A002513, A035444, A082303, A100853, A107034, A318026, A318028.
Sequence in context: A226635 A024788 A285472 * A004101 A003405 A153918
Adjacent sequences: A318024 A318025 A318026 * A318028 A318029 A318030
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KEYWORD
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nonn
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AUTHOR
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Ilya Gutkovskiy, Aug 13 2018
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STATUS
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approved
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