cp's OEIS Frontend

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

%I A000990 M2462 N0978 #88 Aug 23 2025 12:30:23
%S A000990 1,1,3,5,10,16,29,45,75,115,181,271,413,605,895,1291,1866,2648,3760,
%T A000990 5260,7352,10160,14008,19140,26085,35277,47575,63753,85175,113175,
%U A000990 149938,197686,259891,340225,444135,577593,749131,968281,1248320,1604340,2056809,2629357,3353404
%N A000990 Number of plane partitions of n with at most two rows.
%C A000990 Equals row sums of triangle A147767. - _Gary W. Adamson_, Nov 11 2008
%C A000990 Also number of partitions of n into parts of 2 kinds except for 1. - _Reinhard Zumkeller_, Nov 06 2012
%C A000990 Antidiagonal sums of triangle A093010.
%D A000990 G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 105.
%D A000990 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).
%D A000990 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000990 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000990 Alois P. Heinz, <a href="/A000990/b000990.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 1001..7000 from Vaclav Kotesovec)
%H A000990 M. S. Cheema and B. Gordon, <a href="http://dx.doi.org/10.1215/S0012-7094-64-03125-4">Some remarks on two- and three-line partitions</a>, Duke Math. J., 31 (1964), 267-273.
%H A000990 M. S. Cheema, <a href="/A000990/a000990.pdf">Letter to N. J. A. Sloane</a>, Jul 15 1970 [scanned copy]
%H A000990 P. A. MacMahon, <a href="https://archive.org/stream/messengerofmathe52cambuoft#page/112/mode/2up">The connexion between the sum of the squares of the divisors and the number of partitions of a given number</a>, Messenger Math., 54 (1924), 113-116.
%H A000990 R. Newton and A. R. Camacho, <a href="https://arxiv.org/abs/1509.08069">Strangely dual orbifold equivalence I</a>, arXiv preprint arXiv:1509.08069 [math.QA], 2015.
%H A000990 Steven Rayan, <a href="https://arxiv.org/abs/1809.05732">Aspects of the topology and combinatorics of Higgs bundle moduli spaces</a>, arXiv:1809.05732 [math.AG], 2018.
%H A000990 A. G. Shannon, B. Kuloğlu, and E. Özkan, <a href="https://doi.org/10.1007/s40314-025-03179-x">Rhaly terraced sequences their generalizations, properties and applications</a>, Comp. Appl. Math. 44, 226 (2025). See p. 2.
%F A000990 G.f.: 1 / ( (1-x) * Product_{m>=2} (1-x^m)^2 ) = (1-x) / Product_{m>=1} (1-x^m)^2.
%F A000990 G.f.: exp( Sum_{n>=1} ((1+x^n)/(1-x^n))*x^n/n ). - _Paul D. Hanna_, Apr 22 2010
%F A000990 For n>=1, a(n) = A000712(n) - A000712(n-1). - _Vaclav Kotesovec_, Oct 28 2015
%F A000990 a(n) ~ Pi * exp(2*Pi*sqrt(n/3)) / (4 * 3^(5/4) * n^(7/4)). - _Vaclav Kotesovec_, Oct 28 2015
%F A000990 G.f.: exp(Sum_{k>=1} (2*sigma_1(k) - 1)*x^k/k). - _Ilya Gutkovskiy_, Aug 21 2018
%p A000990 b:= proc(n,i) option remember; `if`(n=0, 1,
%p A000990       `if`(i<1, 0, add(binomial(min(i, 2)+j-1, j)*
%p A000990        b(n-i*j, i-1), j=0..n/i)))
%p A000990     end:
%p A000990 a:= n-> b(n$2):
%p A000990 seq(a(n), n=0..45);  # _Alois P. Heinz_, Mar 15 2014
%t A000990 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_ *)
%t A000990 Flatten[{1, Differences[Table[Sum[PartitionsP[j]*PartitionsP[n-j], {j, 0, n}], {n, 0, 50}]]}] (* _Vaclav Kotesovec_, Oct 28 2015 *)
%t A000990 CoefficientList[(1-q)/QPochhammer[q]^2+O[q]^50, q] (* _Jean-François Alcover_, Nov 27 2015 *)
%o A000990 (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 */
%o A000990 (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
%o A000990 (PARI) x='x+O('x^66); Vec((1-x)/eta(x)^2) \\ _Joerg Arndt_, May 01 2013
%o A000990 (Haskell)
%o A000990 a000990 = p $ tail a008619_list where
%o A000990    p _          0 = 1
%o A000990    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A000990 -- _Reinhard Zumkeller_, Nov 06 2012
%o A000990 (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(&*[1-x^j: j in [1..2*m]] )^2 )); // _G. C. Greubel_, Dec 06 2018
%o A000990 (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
%Y A000990 A row of the array in A242641.
%Y A000990 Cf. A147767. - _Gary W. Adamson_, Nov 11 2008
%Y A000990 Cf. A008619, A000070, A000712.
%Y A000990 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).
%K A000990 nonn,easy,changed
%O A000990 0,3
%A A000990 _N. J. A. Sloane_