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A023003
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Number of partitions of n into parts of 4 kinds.
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11
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1, 4, 14, 40, 105, 252, 574, 1240, 2580, 5180, 10108, 19208, 35693, 64960, 116090, 203984, 353017, 602348, 1014580, 1688400, 2778517, 4524760, 7296752, 11658920, 18468245, 29015700, 45235414, 70005376, 107585845, 164245380, 249162620, 375704920, 563251038
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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G.f.: Product_{m>=1} 1/(1-x^m)^4.
a(0)=1, a(n) = (1/n) * Sum_{k=0..n-1} 4*a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
a(n) ~ exp(2 * Pi * sqrt(2*n/3)) / (2^(7/4) * 3^(5/4) * n^(7/4)) * (1 - (35*sqrt(3)/(16*Pi) + Pi/(3*sqrt(3))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
The even bisection of the g.f. A(x) is (A(x) + A(-x))/2 = 1 + 14*x^2 + 105*x^4 + 574*x^6 + ... = Product_{n >= 1} (1 + x^(2*n))^14 / (1 - x^(8*n))^4 = F(x^2)*A(x^8), where F(x) = Product_{n >= 1} (1 + x^n)^14 is the g.f. of A022579.
The odd bisection of the g.f. is (A(x) - A(-x))/2 = 4*x + 40*x^3 + 252*x^5 + 1240*x^7 + ... = (4*x) * Product_{n >= 1} (1 + x^(2*n))^2 * (1 - x^(8*n))^4 / (1 - x^(2*n))^8. (End)
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MAPLE
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with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*4, d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Oct 17 2008
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MATHEMATICA
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nmax=50; CoefficientList[Series[Product[1/(1-x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 28 2015 *)
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PROG
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(PARI) \ps100
for(n=0, 100, print1((polcoeff(1/eta(x)^4, n, x)), ", "))
(Julia) # DedekindEta is defined in A000594.
A023003List(len) = DedekindEta(len, -4)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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