A157209 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = k if k <= floor(n/2) otherwise n-k, and m = 3, read by rows.

1, 1, 1, 1, 11, 1, 1, 54, 54, 1, 1, 229, 822, 229, 1, 1, 932, 8368, 8368, 932, 1, 1, 3747, 72066, 174758, 72066, 3747, 1, 1, 15010, 570006, 2759750, 2759750, 570006, 15010, 1, 1, 60065, 4297714, 37366190, 73850596, 37366190, 4297714, 60065, 1, 1, 240288, 31495488, 460448520, 1591033788, 1591033788, 460448520, 31495488, 240288, 1

Offset

0,5

Links

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

Formula

T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = k if k <= floor(n/2) otherwise n-k, and m = 3.

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

Example

Triangle begins as:
  1;
  1,     1;
  1,    11,       1;
  1,    54,      54,        1;
  1,   229,     822,      229,        1;
  1,   932,    8368,     8368,      932,        1;
  1,  3747,   72066,   174758,    72066,     3747,       1;
  1, 15010,  570006,  2759750,  2759750,   570006,   15010,     1;
  1, 60065, 4297714, 37366190, 73850596, 37366190, 4297714, 60065, 1;

Mathematica

f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1, k-1, m] + (m*k+1)*T[n-1, k, m] + m*f[n, k]*T[n-2, k-1, m]];
Table[T[n, k, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *)

Prog

(Sage)
def f(n, k): return k if (k <= n//2) else n-k
@CachedFunction
def T(n, k, m):  # A157209
    if (k==0 or k==n): return 1
    else: return (m*(n-k) +1)*T(n-1, k-1, m) + (m*k+1)*T(n-1, k, m) + m*f(n, k)*T(n-2, k-1, m)
flatten([[T(n, k, 3) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 10 2022

Crossrefs

Cf. A007318 (m=0), A157207 (m=1), A157208 (m=2), this sequence (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

Sequence in context: A202870 A202872 A144440 * A014469 A157149 A342889

Adjacent sequences: A157206 A157207 A157208 * A157210 A157211 A157212

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 25 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved