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A000990 Number of plane partitions of n with at most two rows.
(Formerly M2462 N0978)
18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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, 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).

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.

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]

R. Newton, A. R. Camacho, Strangely dual orbifold equivalence I, arXiv preprint arXiv:1509.08069 [math.QA], 2015.

FORMULA

G.f.: 1 / ( (1-x) * prod(m>=2, 1-x^m )^2 ) = (1-x) / prod(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

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 *)

PROG

(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

(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

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

CROSSREFS

A row of the array in A242641.

Cf. A147767 [From 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).

Sequence in context: A253769 A070559 A320788 * A129361 A062773 A319130

Adjacent sequences:  A000987 A000988 A000989 * A000991 A000992 A000993

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 21 11:01 EST 2018. Contains 317447 sequences. (Running on oeis4.)