A168295 Triangle T(n, k) = coefficients of (p(x,n)), where p(x, n) = (n-1)! * Sum_{j=1..n} A142458(n, j)*binomial(x+j-1, n-1), read by rows.

1, 1, 2, 2, 10, 10, 6, 52, 120, 80, 24, 280, 1160, 1760, 880, 120, 1520, 10000, 27200, 30800, 12320, 720, 11280, 78160, 343200, 695200, 628320, 209440, 5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800, 40320, 1438080, 15532160, 48294400, 170755200, 445688320, 598160640, 385369600, 96342400

Offset: 1

Author

Roger L. Bagula, Nov 22 2009

Keywords

• sign
• tabl

Examples

			Triangle begins as:
     1;
     1,      2;
     2,     10,     10;
     6,     52,    120,      80;
    24,    280,   1160,    1760,      880;
   120,   1520,  10000,   27200,    30800,    12320;
   720,  11280,  78160,  343200,   695200,   628320,   209440;
  5040, 164640, 784000, 3684800, 12073600, 19490240, 14660800, 4188800;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Crossrefs

Cf. A008544, A142458, A167786.

Programs

Mathematica

T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k-m+1)*T[n-1, k, m]];
A142458[n_, k_]:= A142458[n, k]= T[n,k,3];
p[x_, n_]:= p[x, n]= Sum[A142458[n,k]*Pochhammer[x+k-n+1, n-1], {k, n}];
Table[CoefficientList[p[x, n], x], {n, 1, 12}]//Flatten (* modified by G. C. Greubel, Mar 17 2022 *)

Sage

@CachedFunction
def T(n,k,m): # A142458
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A142458(n,k): return T(n,k,3)
@CachedFunction
def p(n,x): return sum( A142458(n,j)*rising_factorial(x+j-n+1, n-1) for j in (1..n))
def A168295(n,k): return ( p(n,x) ).series(x,n+1).list()[k-1]
flatten([[ A168295(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022

Formula

T(n, k) = coefficients of (p(x,n)), where p(x, n) = (n-1)! * Sum_{j=1..n} A142458(n, j)*binomial(x+j-1, n-1).

From G. C. Greubel, Mar 17 2022: (Start)

T(n, k) = coefficients of (p(n, x)), where p(n, x) = Sum_{j=1..n} A142458(n, j)*Pochhammer(x+j-n+1, n-1).

T(n, 1) = (n-1)!.

T(n, n) = A008544(n-1). (End)

Extensions

Edited by G. C. Greubel, Mar 17 2022