A273596 For n >= 2, a(n) is the number of slim rectangular diagrams of length n.

1, 3, 9, 32, 139, 729, 4515, 32336, 263205, 2401183, 24275037, 269426592, 3257394143, 42615550453, 599875100487, 9040742057760, 145251748024649, 2478320458476795, 44755020000606961, 852823700470009056, 17101229029400788083, 359978633317886558801, 7936631162022905081707

Offset

2,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 2..400

P. Bala, Notes on A273596

Gábor Czédli, Tamás Dékány, Gergő Gyenizse, and Júlia Kulin, The number of slim rectangular lattices, Algebra Universalis, 2016, Volume 75, Issue 1, pp 33-50.

Formula

a(n) = Sum_{1<=r,s; r+s<=n} binomial(n-r-1, s-1) * binomial(n-s-1, r-1) * (n-r-s)!.

a(n) ~ exp(2) * n! / n^2. - Vaclav Kotesovec, Jun 29 2016

a(n) = Sum_{k=0..n} hypergeom([k+1, k-n], [], -1). - Peter Luschny, Oct 05 2017

From Peter Bala, Jan 08 2018: (Start)

a(n) = Sum_{k = 0..n-2} k!*binomial(n+k-1, 2*k+1).

a(n) = (n - 2)*a(n-1) + a(n-2) + 2, with a(2) = 1, a(3) = 3.

a(n+2) = 1/n!*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)* A000522(n)^2.

Row sums of array A143409 read as a triangle.

O.g.f.: Sum_{n >= 0} n!*x^(n+2)/(1 - x)^(2*n+2). Cf. A000179, A000271, A000904 and A127548.

O.g.f. with offset 0: 1/(1 - x) o 1/(1 - x) = 1 + 3*x + 9*x^2 + 32*x^3 + ..., where o denotes the white diamond multiplication of power series. See the Bala link for details. (End)

Example

The initial term is the diagram of the four element diamond shape lattice.

Maple

A273596 := proc (n) option remember; `if`(n = 2, 1, `if`(n = 3, 3, (n-2)*procname(n-1) + procname(n-2) + 2)) end: seq(A273596(n), n = 2..20); # Peter Bala, Jan 08 2017

Mathematica

x = 15;
SRectD = Table[0, {x}];
For[n = 2, n < x, n++,
For[a = 1, a < n, a++,
   For[b = 1, b <= n - a, b++,
    SRectD[[n]] +=
      Binomial[n - a - 1, b - 1]*
       Binomial[n - b - 1, a - 1]*(n - a - b)!;
    ]
   ]
  Print[n, " ", SRectD[[n]]]
]
(* Alternatively: *)
T[n_, k_] := HypergeometricPFQ[{k+1, k-n}, {}, -1];
Table[Sum[T[n, k], {k, 0, n}], {n, 0, 22}] (* Peter Luschny, Oct 05 2017 *)

Prog

(PARI) a(n)= sum(rps=1, n, sum(r=1, n, s = rps-r; binomial(n-r-1, s-1) * binomial(n-s-1, r-1) * (n-r-s)!)); \\ Michel Marcus, Jun 12 2016

Crossrefs

Cf. A273988, A000522, A143409, A000179, A000271, A000904, A127548.

Partial sums of A155857.

Sequence in context: A129416 A210689 A334642 * A009356 A341302 A297209

Adjacent sequences: A273593 A273594 A273595 * A273597 A273598 A273599

Keyword

nonn,easy

Author

Tamas Dekany, May 26 2016

Status

approved