A273596 - OEIS

%I #48 Jun 26 2022 08:33:40

%S 1,3,9,32,139,729,4515,32336,263205,2401183,24275037,269426592,

%T 3257394143,42615550453,599875100487,9040742057760,145251748024649,

%U 2478320458476795,44755020000606961,852823700470009056,17101229029400788083,359978633317886558801,7936631162022905081707

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

%H Vaclav Kotesovec, <a href="/A273596/b273596.txt">Table of n, a(n) for n = 2..400</a>

%H P. Bala, <a href="/A273596/a273596.pdf">Notes on A273596</a>

%H Gábor Czédli, Tamás Dékány, Gergő Gyenizse, and Júlia Kulin, <a href="http://dx.doi.org/10.1007/s00012-015-0363-y">The number of slim rectangular lattices</a>, Algebra Universalis, 2016, Volume 75, Issue 1, pp 33-50.

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

%F a(n) ~ exp(2) * n! / n^2. - _Vaclav Kotesovec_, Jun 29 2016

%F a(n) = Sum_{k=0..n} hypergeom([k+1, k-n], [], -1). - _Peter Luschny_, Oct 05 2017

%F From _Peter Bala_, Jan 08 2018: (Start)

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

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

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

%F Row sums of array A143409 read as a triangle.

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

%F 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)

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

%p 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

%t x = 15;

%t SRectD = Table[0, {x}];

%t For[n = 2, n < x, n++,

%t For[a = 1, a < n, a++,

%t    For[b = 1, b <= n - a, b++,

%t     SRectD[[n]] +=

%t       Binomial[n - a - 1, b - 1]*

%t        Binomial[n - b - 1, a - 1]*(n - a - b)!;

%t     ]

%t    ]

%t   Print[n, " ", SRectD[[n]]]

%t ]

%t (* Alternatively: *)

%t T[n_, k_] := HypergeometricPFQ[{k+1, k-n}, {}, -1];

%t Table[Sum[T[n,k], {k,0,n}], {n,0,22}] (* _Peter Luschny_, Oct 05 2017 *)

%o (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

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

%Y Partial sums of A155857.

%K nonn,easy

%O 2,2

%A _Tamas Dekany_, May 26 2016