A000990 Number of plane partitions of n with at most two rows.

1, 1, 3, 5, 10, 16, 29, 45, 75, 115, 181, 271, 413, 605, 895, 1291, 1866, 2648, 3760, 5260, 7352, 10160, 14008, 19140, 26085, 35277, 47575, 63753, 85175, 113175, 149938, 197686, 259891, 340225, 444135, 577593, 749131, 968281, 1248320, 1604340, 2056809, 2629357, 3353404

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

Comments

Equals row sums of triangle A147767. - Gary W. Adamson, Nov 11 2008

Also number of partitions of n into parts of 2 kinds except for 1. - Reinhard Zumkeller, Nov 06 2012

Antidiagonal sums of triangle A093010.

References

• G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 105.
• 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. (1.7).
• 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, Table of n, a(n) for n = 0..10000 (terms n = 1001..7000 from Vaclav Kotesovec)
• M. S. Cheema and B. Gordon, Some remarks on two- and three-line partitions, Duke Math. J., 31 (1964), 267-273.
• M. S. Cheema, Letter to N. J. A. Sloane, Jul 15 1970 [scanned copy]
• 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.
• R. Newton and A. R. Camacho, Strangely dual orbifold equivalence I, arXiv preprint arXiv:1509.08069 [math.QA], 2015.
• Steven Rayan, Aspects of the topology and combinatorics of Higgs bundle moduli spaces, arXiv:1809.05732 [math.AG], 2018.
• A. G. Shannon, B. Kuloğlu, and E. Özkan, Rhaly terraced sequences their generalizations, properties and applications, Comp. Appl. Math. 44, 226 (2025). See p. 2.

Crossrefs

A row of the array in A242641.

Cf. A147767. - Gary W. Adamson, Nov 11 2008

Cf. A008619, A000070, A000712.

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).

Programs

Haskell

a000990 = p $ tail a008619_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 06 2012

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(&*[1-x^j: j in [1..2*m]] )^2 )); // G. C. Greubel, Dec 06 2018

Maple

b:= proc(n,i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(min(i, 2)+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, 2]+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 17 2014, after Alois P. Heinz *)
Flatten[{1, Differences[Table[Sum[PartitionsP[j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 50}]]}] (* Vaclav Kotesovec, Oct 28 2015 *)
CoefficientList[(1-q)/QPochhammer[q]^2+O[q]^50, q] (* Jean-François Alcover, Nov 27 2015 *)

PARI

a(n)=if(n<0,0,polcoeff((1-x)/prod(k=1,n,1-x^k,1+x*O(x^n))^2,n)) /* Michael Somos, Jan 29 2005 */

PARI

{a(n)=polcoeff(exp(sum(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))*x^m/m)),n)} \\ Paul D. Hanna, Apr 22 2010

PARI

x='x+O('x^66); Vec((1-x)/eta(x)^2) \\ Joerg Arndt, May 01 2013

Sage

s=((1-x)/prod(1-x^j for j in (1..60))^2).series(x, 50); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 06 2018

Formula

G.f.: 1 / ( (1-x) * Product_{m>=2} (1-x^m)^2 ) = (1-x) / Product_{m>=1} (1-x^m)^2.

G.f.: exp( Sum_{n>=1} ((1+x^n)/(1-x^n))*x^n/n ). - Paul D. Hanna, Apr 22 2010

For n>=1, a(n) = A000712(n) - A000712(n-1). - Vaclav Kotesovec, Oct 28 2015

a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Oct 28 2015

G.f.: exp(Sum_{k>=1} (2*sigma_1(k) - 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018