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A156764 Triangle T(n, k, m) = b(n, m)/(b(k, m)*b(n-k, m)), with T(0, k, m) = 1, b(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (-1)^(j+i)*(j+1)*(k+1)^i*StirlingS1(j-1, i) ), b(n, 0) = n!, and m = 3, read by rows. 2
1, 1, 1, 1, 6, 1, 1, 40, 40, 1, 1, 300, 2000, 300, 1, 1, 2520, 126000, 126000, 2520, 1, 1, 23520, 9878400, 74088000, 9878400, 23520, 1, 1, 241920, 948326400, 59744563200, 59744563200, 948326400, 241920, 1, 1, 2721600, 109734912000, 64524128256000, 542002677350400, 64524128256000, 109734912000, 2721600, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

FORMULA

T(n, k, m) = b(n, m)/(b(k, m)*b(n-k, m)), with T(0, k, m) = 1, b(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} (-1)^(j+i)*(j+1)*(k+1)^i*StirlingS1(j-1, i) ), b(n, 0) = n!, and m = 3.

T(n, k, m) = f(n, m)/(f(k, m)*f(n-k, m)), with T(0, k, m) = 1, f(n, k) = (-1)^n*(n + 1)!*BarnesG(n+k+1)/(Gamma(k+1)^n*BarnesG(k+1)), f(n, 0) = n!, and m = 3. - G. C. Greubel, Jun 20 2021

EXAMPLE

Triangle begins as:

1;

1, 1;

1, 6, 1;

1, 40, 40, 1;

1, 300, 2000, 300, 1;

1, 2520, 126000, 126000, 2520, 1;

1, 23520, 9878400, 74088000, 9878400, 23520, 1;

1, 241920, 948326400, 59744563200, 59744563200, 948326400, 241920, 1;

MATHEMATICA

(* First program *)

b[n_, k_]:= If[k==0, n!, Product[Sum[(-1)^(i+j)*(j+1)*StirlingS1[j-1, i]*(k+1)^i, {i, 0, j-1}], {j, 1, n}]];

T[n_, k_, m_] = If[n==0, 1, b[n, m]/(b[k, m]*b[n-k, m])];

Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 20 2021 *)

(* Second program *)

f[n_, k_]:= If[k==0, n!, (-1)^n*(n+1)!*BarnesG[n+k+1]/(Gamma[k+1]^n*BarnesG[k+1])];

T[n_, k_, m_]:= If[n==0, 1, f[n, m]/(f[k, m]*f[n-k, m])];

Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 20 2021 *)

PROG

(Sage)

def f(n, k): return factorial(n) if (k==0) else (-1)^n*factorial(n+1)*product( rising_factorial(k+1, j) for j in (0..n-1) )

def T(n, k, m): return 1 if (n==0) else f(n, m)/(f(k, m)*f(n-k, m))

flatten([[T(n, k, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 20 2021

CROSSREFS

Cf. A007318 (m=0), A156584 (m=1), this sequence (m=3).

Sequence in context: A158116 A172343 A058875 * A156765 A015117 A287020

Adjacent sequences: A156761 A156762 A156763 * A156765 A156766 A156767

KEYWORD

nonn,tabl

AUTHOR

Roger L. Bagula, Feb 15 2009

EXTENSIONS

Edited by G. C. Greubel, Jun 20 2021

STATUS

approved

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Last modified December 2 12:19 EST 2022. Contains 358493 sequences. (Running on oeis4.)