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A000991
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Number of 3-line partitions of n.
(Formerly M2554 N1011)
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13
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1, 1, 3, 6, 12, 21, 40, 67, 117, 193, 319, 510, 818, 1274, 1983, 3032, 4610, 6915, 10324, 15235, 22371, 32554, 47119, 67689, 96763, 137404, 194211, 272939, 381872, 531576, 736923, 1016904, 1397853, 1913561, 2610023, 3546507, 4802694, 6481101, 8718309, 11689929, 15627591, 20828892
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OFFSET
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0,3
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COMMENTS
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Planar partitions into at most three rows. - Joerg Arndt, May 01 2013
Number of partitions of n where there is one sort of part 1, two sorts of part 2, and three sorts of every other part. - Joerg Arndt, Mar 15 2014
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REFERENCES
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L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145, eq. (1.8).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: (1-x)^2 * (1-x^2) / Product_(k>=1, 1-x^k )^3.
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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, 3)+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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b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[Min[i, 3]+j-1, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)
nmax = 40; CoefficientList[Series[(1-x)^2 * (1-x^2) * Product[1/(1-x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 28 2015 *)
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PROG
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(PARI) x='x+O('x^66); Vec((1-x)^2*(1-x^2)/eta(x)^3) \\ Joerg Arndt, May 01 2013
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)^2*(1-x^2)/(&*[1-x^j: j in [1..2*m]])^3 )); // G. C. Greubel, Dec 06 2018
(Sage)
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(50)
s = (1-x)^2 * (1-x^2) / prod(1-x^j for j in (1..60))^3
s.coefficients()
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Prepended a(0)=1, added more terms, Joerg Arndt, May 01 2013
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STATUS
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approved
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