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.
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
Offset: 0
Examples
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;
Links
- G. C. Greubel, Rows n = 0..30 of the triangle, flattened
Programs
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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 *) -
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
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
Extensions
Edited by G. C. Greubel, Jun 20 2021