A155493 Triangle T(n, k) = binomial(n+1, k)*A142459(n+1, k+1)/(k+1), read by rows.

1, 1, 1, 1, 15, 1, 1, 118, 118, 1, 1, 770, 3540, 770, 1, 1, 4671, 67810, 67810, 4671, 1, 1, 27321, 1039689, 3085355, 1039689, 27321, 1, 1, 156220, 14006244, 99524810, 99524810, 14006244, 156220, 1, 1, 878868, 173788752, 2602528824, 6090918372, 2602528824, 173788752, 878868, 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 = 4.

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

T(n, k) = binomial(n+1, k)*A142459(n+1, k+1)/(k+1).

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

Example

Triangle begins as:
  1;
  1,      1;
  1,     15,         1;
  1,    118,       118,          1;
  1,    770,      3540,        770,          1;
  1,   4671,     67810,      67810,       4671,          1;
  1,  27321,   1039689,    3085355,    1039689,      27321,         1;
  1, 156220,  14006244,   99524810,   99524810,   14006244,    156220,      1;
  1, 878868, 173788752, 2602528824, 6090918372, 2602528824, 173788752, 878868, 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, 4], {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, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 01 2022

Crossrefs

Cf. A001263 (m=0), A155467 (m=1), A155491 (m=3), this sequence (m=4).

Cf. A142459.

Sequence in context: A111805 A238754 A176226 * A156939 A174187 A174693

Adjacent sequences: A155490 A155491 A155492 * A155494 A155495 A155496

Keyword

nonn,tabl

Author

Roger L. Bagula, Jan 23 2009

Extensions

Edited by G. C. Greubel, Apr 01 2022

Status

approved