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

1,3

Links

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A008544, A142458, A167786.

Sequence in context: A135996 A141610 A019241 * A309751 A249152 A216708

Adjacent sequences: A168292 A168293 A168294 * A168296 A168297 A168298

Keyword

sign,tabl

Author

Roger L. Bagula, Nov 22 2009

Extensions

Edited by G. C. Greubel, Mar 17 2022

Status

approved