cp's OEIS Frontend

A006330 Number of corners, or planar partitions of n with only one row and one column.

%I A006330 M2553 #56 Nov 07 2017 18:15:58
%S A006330 1,1,3,6,12,21,38,63,106,170,272,422,653,986,1482,2191,3218,4666,6726,
%T A006330 9592,13602,19122,26733,37102,51232,70292,95989,130356,176246,237120,
%U A006330 317724,423840,563266,745562,983384,1292333,1692790,2209886,2876132
%N A006330 Number of corners, or planar partitions of n with only one row and one column.
%C A006330 The first four terms a(0), a(1), a(2), a(3) agree with sequence A000219 for unrestricted planar partitions, since the restriction does not rule anything out. For a(4) there is just one planar partition which doesn't satisfy the restriction, four 1's arranged in a square. So A000219 has fifth term 13 and here we have 12.
%C A006330 a(n) + A001523(n) = A000712(n). - _Michael Somos_, Jul 22 2003
%C A006330 Number of unimodal compositions of n+1 where the maximal part appears once, see example. [_Joerg Arndt_, Jun 11 2013]
%D A006330 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006330 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see page 77.
%H A006330 Alois P. Heinz, <a href="/A006330/b006330.txt">Table of n, a(n) for n = 0..10000</a>
%H A006330 G. E. Andrews, <a href="http://dx.doi.org/10.1090/S0273-0979-07-01180-9">Euler's "De Partitio Numerorum"</a>, Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See (5.6).
%H A006330 F. C. Auluck, <a href="http://dx.doi.org/10.1017/S0305004100027134">On some new types of partitions associated with generalized Ferrers graphs</a>, Proc. Cambridge Philos. Soc. 47, (1951), 679-686.
%H A006330 Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, <a href="https://arxiv.org/abs/1606.07070">Twisted sectors from plane partitions</a>, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1.
%H A006330 G. Kreweras, <a href="http://dx.doi.org/10.1016/0012-365X(79)90163-8">Sur les extensions linéaires d'une famille particulière d'ordres partiels</a>, Discrete Math., 27 (1979), 279-295.
%H A006330 G. Kreweras, <a href="/A006330/a006330_1.pdf">Sur les extensions linéaires d'une famille particulière d'ordres partiels</a>, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy)
%H A006330 G. Kreweras, <a href="/A006330/a006330.pdf">Letter to N. J. A. Sloane</a>
%F A006330 G.f.: 1+Sum_{k>0} x^k/(Product_{i=1..k} (1-x^i))^2.
%F A006330 G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1 - x^k)^2. - _Michael Somos_, Jul 28 2003
%F A006330 Convolution product of A197870 and A000712. - _Michael Somos_, Feb 22 2015
%F A006330 a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - _Vaclav Kotesovec_, Jun 22 2015
%e A006330 From _Joerg Arndt_, Jun 11 2013: (Start)
%e A006330 There are a(4)=12 unimodal compositions of 4+1=5 where the maximal part appears once:
%e A006330 01:  [ 1 1 1 2 ]
%e A006330 02:  [ 1 1 2 1 ]
%e A006330 03:  [ 1 1 3 ]
%e A006330 04:  [ 1 2 1 1 ]
%e A006330 05:  [ 1 3 1 ]
%e A006330 06:  [ 1 4 ]
%e A006330 07:  [ 2 1 1 1 ]
%e A006330 08:  [ 2 3 ]
%e A006330 09:  [ 3 1 1 ]
%e A006330 10:  [ 3 2 ]
%e A006330 11:  [ 4 1 ]
%e A006330 12:  [ 5 ]
%e A006330 (End)
%e A006330 G.f. = 1 + x + 3*x^2 + 6*x^3 + 12*x^4 + 21*x^5 + 38*x^6 + 63*x^7 + 106*x^8 + ...
%t A006330 a[0] = 1; a[n_] := SeriesCoefficient[ Sum[x^k/Product[1 - x^i, {i, 1, k}]^2, {k, 1, n}] + 1, {x, 0, n}]; Array[a, 39, 0] (* _Jean-François Alcover_, Mar 13 2014 *)
%o A006330 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / prod(i=1, k, 1 - x^i, 1 + x*O(x^n))^2, 1), n))};
%o A006330 (PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(1 + 8*n) - 1)\2, (-1)^k * x^((k + k^2)/2)) / eta(x + x*O(x^n))^2, n))};
%Y A006330 Cf. A000219, A000712, A197870.
%Y A006330 Column k=1 of A247255.
%Y A006330 Row sums of A259100.
%K A006330 nonn
%O A006330 0,3
%A A006330 _N. J. A. Sloane_
%E A006330 Edited and extended by _Moshe Shmuel Newman_, Jun 10 2003