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
Examples
The initial term is the diagram of the four element diamond shape lattice.
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.
Crossrefs
Programs
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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
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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 *) -
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
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. 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)