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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

Alois P. Heinz and 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.

P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116.

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-Fran├žois 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) s=((1-x)^2*(1-x^2)/prod(1-x^j for j in (1..60))^3).series(x, 50); s.coefficients(x, sparse=False) # 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 * A095093 A280473 A139422

Adjacent sequences:  A000988 A000989 A000990 * A000992 A000993 A000994

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

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 October 15 01:40 EDT 2019. Contains 328025 sequences. (Running on oeis4.)