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

0,5

Links

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

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)

Example

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;

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 *)

Prog

(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

Crossrefs

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

Cf. A142458.

Sequence in context: A174672 A174151 A342890 * A142460 A156280 A166962

Adjacent sequences: A155488 A155489 A155490 * A155492 A155493 A155494

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 23 2009

Extensions

Edited by G. C. Greubel, Apr 01 2022

Status

approved