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 A006330 Number of corners, or planar partitions of n with only one row and one column. (Formerly M2553) 14
 1, 1, 3, 6, 12, 21, 38, 63, 106, 170, 272, 422, 653, 986, 1482, 2191, 3218, 4666, 6726, 9592, 13602, 19122, 26733, 37102, 51232, 70292, 95989, 130356, 176246, 237120, 317724, 423840, 563266, 745562, 983384, 1292333, 1692790, 2209886, 2876132 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. a(n) + A001523(n) = A000712(n). - Michael Somos, Jul 22 2003 Number of unimodal compositions of n+1 where the maximal part appears once, see example. [Joerg Arndt, Jun 11 2013] REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see page 77. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573. See (5.6). F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47, (1951), 679-686. Shouvik Datta, M. R. Gaberdiel, W. Li, C. Peng, Twisted sectors from plane partitions, arXiv preprint arXiv:1606.07070 [hep-th], 2016. See Sect. 2.1. G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. G. Kreweras, Sur les extensions linéaires d'une famille particulière d'ordres partiels, Discrete Math., 27 (1979), 279-295. (Annotated scanned copy) G. Kreweras, Letter to N. J. A. Sloane FORMULA G.f.: 1+Sum_{k>0} x^k/(Product_{i=1..k} (1-x^i))^2. G.f.: (Sum_{k>=0} (-1)^k * x^(k(k+1)/2)) / (Product_{k>0} 1 - x^k)^2. - Michael Somos, Jul 28 2003 Convolution product of A197870 and A000712. - Michael Somos, Feb 22 2015 a(n) ~ exp(2*Pi*sqrt(n/3)) / (8 * 3^(3/4) * n^(5/4)) [Auluck, 1951]. - Vaclav Kotesovec, Jun 22 2015 EXAMPLE From Joerg Arndt, Jun 11 2013: (Start) There are a(4)=12 unimodal compositions of 4+1=5 where the maximal part appears once: 01:  [ 1 1 1 2 ] 02:  [ 1 1 2 1 ] 03:  [ 1 1 3 ] 04:  [ 1 2 1 1 ] 05:  [ 1 3 1 ] 06:  [ 1 4 ] 07:  [ 2 1 1 1 ] 08:  [ 2 3 ] 09:  [ 3 1 1 ] 10:  [ 3 2 ] 11:  [ 4 1 ] 12:  [ 5 ] (End) 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 + ... MATHEMATICA a = 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 *) PROG (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))}; (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))}; CROSSREFS Cf. A000219, A000712, A197870. Column k=1 of A247255. Row sums of A259100. Sequence in context: A162920 A247662 A215005 * A293636 A087503 A092176 Adjacent sequences:  A006327 A006328 A006329 * A006331 A006332 A006333 KEYWORD nonn AUTHOR EXTENSIONS Edited and extended by Moshe Shmuel Newman, Jun 10 2003 STATUS approved

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Last modified December 11 01:07 EST 2019. Contains 329910 sequences. (Running on oeis4.)