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A273596
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For n >= 2, a(n) is the number of slim rectangular diagrams of length n.
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3
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1, 3, 9, 32, 139, 729, 4515, 32336, 263205, 2401183, 24275037, 269426592, 3257394143, 42615550453, 599875100487, 9040742057760, 145251748024649, 2478320458476795, 44755020000606961, 852823700470009056, 17101229029400788083, 359978633317886558801, 7936631162022905081707
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OFFSET
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2,2
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LINKS
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FORMULA
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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) = Sum_{k=0..n} hypergeom([k+1, k-n], [], -1). - Peter Luschny, Oct 05 2017
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)
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EXAMPLE
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The initial term is the diagram of the four element diamond shape lattice.
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MAPLE
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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
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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