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A174117 Triangle T(n, k) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1, read by rows. 5
1, 1, 1, 1, 3, 1, 1, 8, 8, 1, 1, 15, 40, 15, 1, 1, 24, 120, 120, 24, 1, 1, 35, 280, 525, 280, 35, 1, 1, 48, 560, 1680, 1680, 560, 48, 1, 1, 63, 1008, 4410, 7056, 4410, 1008, 63, 1, 1, 80, 1680, 10080, 23520, 23520, 10080, 1680, 80, 1, 1, 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

FORMULA

Let c(n) = Product_{j=2..n} (j^2 - 1) for n > 1 otherwise 1 then the number triangle is given by T(n, k) = c(n)/(c(k)*c(n-k)).

From G. C. Greubel, Feb 11 2021: (Start)

T(n, k) = (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k) with T(n, 0) = T(n, n) = 1.

T(n, k) = 2*((n+1)*(n-k)/(k+1))*A001263(n, k).

Sum_{k=0..n} T(n, k) = (2/(n+2))*( (n^2-1)*C_{n} + 1), where C_{n} are the Catalan numbers (A000108). (End)

EXAMPLE

Triangle begins as:

  1;

  1,  1;

  1,  3,    1;

  1,  8,    8,     1;

  1, 15,   40,    15,     1;

  1, 24,  120,   120,    24,     1;

  1, 35,  280,   525,   280,    35,     1;

  1, 48,  560,  1680,  1680,   560,    48,     1;

  1, 63, 1008,  4410,  7056,  4410,  1008,    63,    1;

  1, 80, 1680, 10080, 23520, 23520, 10080,  1680,   80,  1;

  1, 99, 2640, 20790, 66528, 97020, 66528, 20790, 2640, 99, 1;

MATHEMATICA

(* First program *)

c[n_]:= If[n<2, 1, Product[i^2 -1, {i, 2, n}]];

T[n_, k_]:= c[n]/(c[k]*c[n-k]);

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

(* Second program *)

T[n_, k_]:= If[k==0 || k==n, 1, (2*k/(k+1))*Binomial[n+1, k]*Binomial[n-1, k]];

Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 11 2021 *)

PROG

(Sage)

def T(n, k): return 1 if (k==0 or k==n) else (2*k/(k+1))*binomial(n-1, k)*binomial(n+1, k)

flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 11 2021

(Magma)

T:= func< n, k | k eq 0 or k eq n select 1 else (2*k/(k+1))*Binomial(n-1, k)*Binomial(n+1, k) >;

[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 11 2021

CROSSREFS

Cf. A174116, A174119, A174124, A174125.

Cf. A000108, A001263.

Sequence in context: A094816 A097712 A238688 * A157210 A034801 A331890

Adjacent sequences:  A174114 A174115 A174116 * A174118 A174119 A174120

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Mar 08 2010

EXTENSIONS

Edited by G. C. Greubel, Feb 11 2021

STATUS

approved

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Last modified August 14 17:54 EDT 2022. Contains 356122 sequences. (Running on oeis4.)