A155491 Triangle T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1), read by rows.

1, 1, 1, 1, 12, 1, 1, 78, 78, 1, 1, 415, 1820, 415, 1, 1, 2031, 27410, 27410, 2031, 1, 1, 9534, 330225, 959350, 330225, 9534, 1, 1, 43660, 3488884, 23935450, 23935450, 3488884, 43660, 1, 1, 196569, 33888576, 484631574, 1120179060, 484631574, 33888576, 196569, 1

Offset: 0

Author

Roger L. Bagula, Jan 23 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     12,        1;
  1,     78,       78,         1;
  1,    415,     1820,       415,          1;
  1,   2031,    27410,     27410,       2031,         1;
  1,   9534,   330225,    959350,     330225,      9534,        1;
  1,  43660,  3488884,  23935450,   23935450,   3488884,    43660,      1;
  1, 196569, 33888576, 484631574, 1120179060, 484631574, 33888576, 196569, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A001263 (m=0), A155467 (m=1), this sequence (m=3), A155493 (m=4).

Cf. A142458.

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]];
T[n_, k_, m_]:= Binomial[n+1,k]*t[n+1,k+1,m]/(k+1);
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Apr 01 2022 *)

Sage

@CachedFunction
def t(n,k,m):
    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 T(n,k,m): return binomial(n+1,k)*t(n+1,k+1,m)/(k+1)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 01 2022

Formula

T(n, k) = binomial(n+1, k)*t(n, k, m)/(k+1), where t(n,k,m) = (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m), t(n,1,m) = t(n,n,m) = 1, and m = 3.

From G. C. Greubel, Apr 01 2022: (Start)

T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1).

T(n, n-k) = T(n, k). (End)

Extensions

Edited by G. C. Greubel, Apr 01 2022