A127548 -id:A127548 - cp's OEIS frontend

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

Tamas Dekany, May 26 2016

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

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.

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

Partial sums of A155857.

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

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 *)

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

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)

A332709 Triangle T(n,k) read by rows where T(n,k) is the number of ménage permutations that map 1 to k, with 3 <= k <= n.

Original entry on oeis.org

1, 1, 1, 4, 5, 4, 20, 20, 20, 20, 115, 116, 117, 116, 115, 787, 791, 791, 791, 791, 787, 6184, 6203, 6204, 6205, 6204, 6203, 6184, 54888, 55000, 55004, 55004, 55004, 55004, 55000, 54888, 542805, 543576, 543595, 543596, 543597, 543596, 543595, 543576, 542805

Offset: 3

Peter Kagey, Feb 20 2020

Rows are palindromic.
Conjecture: Rows are unimodal (i.e., increasing, then decreasing).
Conjecture: T(n,k) - T(n,k-1) = A127548(n-2k+4) for n >= 2k - 3. - Peter Kagey, Jan 22 2021

			Triangle begins:
  n\k|     3      4      5      6      7      8      9     10
  ---+--------------------------------------------------------
   3 |     1
   4 |     1,     1
   5 |     4,     5,     4
   6 |    20,    20,    20,    20
   7 |   115,   116,   117,   116,   115
   8 |   787,   791,   791,   791,   791,   787
   9 |  6184,  6203,  6204,  6205,  6204,  6203,  6184
  10 | 54888, 55000, 55004, 55004, 55004, 55004, 55000, 54888
For n=5 and k=3, the T(5,3)=4 permutations on five letters that start with a 3 are 31524, 34512, 35124, and 35212.

Peter Kagey, Table of n, a(n) for n = 3..1277 (first 50 rows)

Cf. A127548.
First column given by A258664.
Second column given by A258665.
Third column given by A258666.
Fourth column given by A258667.
Row sums given by A000179.

T[n_, k_] :=
Sum[Sum[(-1)^i*(n - i - 1)!*Binomial[2*k - j - 4, j]*
    Binomial[2*(n - k + 1) - i + j, i - j], {j, Max[k + i - n - 1, 0],
     Min[i, k - 2]}], {i, 0, n - 1}]
(* Peter Kagey, Jan 22 2021 *)

T(n,k) = Sum_{i=0..n-1} Sum_{j=max(k+i-n-1,0)..min(i,k-2)} (-1)^i*(n-i-1)! * binomial(2k-j-4,j) * binomial(2(n-k+1)-i+j,i-j).

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

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