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0,3

The first four terms a(0), a(1), a(2), a(3) agree with sequenceA000219 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. SoA000219 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]

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.

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

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 ofA197870 andA000712. -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

FromJoerg 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 + ...

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 *)

(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))};

Cf.A000219,A000712,A197870.

Column k=1 ofA247255.

Row sums ofA259100.

Sequence in context:A247662A337462A215005 *A293636A087503A092176

Adjacent sequences:A006327A006328A006329 *A006331A006332A006333

Edited and extended byMoshe Shmuel Newman, Jun 10 2003

approved