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A000991 Number of 3-line partitions of n.
(Formerly M2554 N1011)
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
REFERENCES
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).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..6000 (first 1000 terms from Alois P. Heinz)
M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
FORMULA
G.f.: (1-x)^2 * (1-x^2) / Product_(k>=1, 1-x^k )^3.
For n>=4, a(n) = A000716(n) - 2*A000716(n-1) + 2*A000716(n-3) - A000716(n-4). - Vaclav Kotesovec, Oct 28 2015
a(n) ~ Pi^3 * exp(Pi*sqrt(2*n)) / (16*n^3). - Vaclav Kotesovec, Oct 28 2015
MAPLE
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):
seq(a(n), n=0..45); # Alois P. Heinz, Mar 15 2014
MATHEMATICA
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 *)
PROG
(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()
# G. C. Greubel, Dec 06 2018
CROSSREFS
A row of the array in A242641.
Sequences "number of r-line partitions": A000041 (r=1), A000990 (r=2), A000991 (r=3), A002799 (r=4), A001452 (r=5), A225196 (r=6), A225197 (r=7), A225198 (r=8), A225199 (r=9).
Sequence in context: A293636 A087503 A092176 * A345028 A095093 A280473
KEYWORD
nonn
AUTHOR
EXTENSIONS
G.f. corrected by Sean A. Irvine, Oct 19 2011
G.f. corrected by Joerg Arndt, May 01 2013
Prepended a(0)=1, added more terms, Joerg Arndt, May 01 2013
STATUS
approved

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Last modified March 19 02:51 EDT 2024. Contains 370952 sequences. (Running on oeis4.)