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A225196
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Number of 6-line partitions of n (i.e., planar partitions of n with at most 6 lines).
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10
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1, 1, 3, 6, 13, 24, 48, 85, 157, 274, 481, 816, 1388, 2298, 3798, 6170, 9968, 15895, 25209, 39550, 61703, 95431, 146757, 224036, 340189, 513233, 770415, 1149933, 1708277, 2524846, 3715285, 5441762, 7937671, 11529512, 16681995, 24043245, 34527521, 49404590, 70452001, 100128249
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OFFSET
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0,3
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COMMENTS
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Number of partitions of n where there are k sorts of parts k for k<=5 and six sorts of all other parts. - Joerg Arndt, Mar 15 2014
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LINKS
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FORMULA
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G.f.: 1/Product_{n>=1} (1-x^n)^min(n,6). - Joerg Arndt, Mar 15 2014
a(n) ~ 2160 * Pi^15 * exp(2*Pi*sqrt(n)) / n^(39/4). - Vaclav Kotesovec, Oct 28 2015
G.f.: (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/( Prod_{j>=1} (1-x^j ) )^6. - G. C. Greubel, Dec 06 2018
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
min(d, 6)*d, d=divisors(j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 6]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Feb 18 2015, Alois P. Heinz *)
m:=50; CoefficientList[Series[(1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/( Product[(1-x^j), {j, 1, m}])^6, {x, 0, m}], x] (* G. C. Greubel, Dec 06 2018 *)
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PROG
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(PARI) x='x+O('x^66); r=6; Vec( prod(k=1, r-1, (1-x^k)^(r-k)) / eta(x)^r )
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/(&*[1-x^j: j in [1..2*m]] )^6 )); // G. C. Greubel, Dec 06 2018
(Sage)
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = (1-x)^5*(1-x^2)^4*(1-x^3)^3*(1-x^4)^2*(1-x^5)/prod(1-x^j for j in (1..60))^6
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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