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A225199
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Number of 9-line partitions of n (i.e., planar partitions of n with at most 9 lines).
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10
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1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 499, 856, 1471, 2466, 4124, 6788, 11110, 17965, 28890, 45995, 72819, 114354, 178577, 276952, 427279, 655199, 999773, 1517388, 2292377, 3446462, 5159352, 7689517, 11414606, 16875813, 24856366, 36474188, 53334376, 77717219, 112874158, 163403202
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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<=8 and nine sorts of all other parts. - Joerg Arndt, Mar 15 2014
In general, "number of r-line partitions" is asymptotic to (Product_{j=1..r-1} j!) * Pi^(r*(r-1)/2) * r^((r^2 + 1)/4) * exp(Pi*sqrt(2*n*r/3)) / (2^((r*(r+2)+5)/4) * 3^((r^2 + 1)/4) * n^((r^2 + 3)/4)). - Vaclav Kotesovec, Oct 28 2015
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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,9). - Joerg Arndt, Mar 15 2014
a(n) ~ 2101805306799541875 * sqrt(3) * Pi^36 * exp(Pi*sqrt(6*n)) / (8*n^21). [The convergence is very slow, numerical verification needs more than 1000000 terms.] - Vaclav Kotesovec, Oct 28 2015
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(binomial(min(i, 9)+j-1, j)*
b(n-i*j, i-1), j=0..n/i)))
end:
a:= n-> b(n$2):
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MATHEMATICA
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a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Min[d, 9]*d, {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)
m:=50; r:=9; CoefficientList[Series[Product[(1-x^k)^(r-k), {k, 1, r-1}]/( Product[(1-x^j), {j, 1, m}])^r, {x, 0, m}], x] (* G. C. Greubel, Dec 10 2018 *)
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PROG
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(PARI) x='x+O('x^66); r=9; Vec( prod(k=1, r-1, (1-x^k)^(r-k)) / eta(x)^r )
(Magma) m:=50; r:=9; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1-x^k)^(r-k): k in [1..r-1]])/(&*[1-x^j: j in [1..2*m]] )^r )); // G. C. Greubel, Dec 10 2018
(Sage)
m=50; r=9
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(m)
s = (prod((1-x^k)^(r-k) for k in (1..r-1))/prod(1-x^j for j in (1..m+2))^r)
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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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