A157278 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) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 3, read by rows.

1, 1, 1, 1, 14, 1, 1, 69, 69, 1, 1, 292, 1134, 292, 1, 1, 1187, 11686, 11686, 1187, 1, 1, 4770, 100737, 254132, 100737, 4770, 1, 1, 19105, 795723, 4061249, 4061249, 795723, 19105, 1, 1, 76448, 5990296, 55157324, 111691642, 55157324, 5990296, 76448, 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) = 2*k if k <= floor(n/2) otherwise 2*(n-k), and m = 3.

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

Example

Triangle begins as:
  1;
  1,     1;
  1,    14,       1;
  1,    69,      69,        1;
  1,   292,    1134,      292,         1;
  1,  1187,   11686,    11686,      1187,        1;
  1,  4770,  100737,   254132,    100737,     4770,       1;
  1, 19105,  795723,  4061249,   4061249,   795723,   19105,     1;
  1, 76448, 5990296, 55157324, 111691642, 55157324, 5990296, 76448, 1;

Mathematica

f[n_, k_]:= If[k<=Floor[n/2], 2*k, 2*(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, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 06 2022 *)

Prog

(Sage)
def f(n, k): return 2*k if (k <= n//2) else 2*(n-k)
@CachedFunction
def T(n, k, m):  # A157278
    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..12)]) # G. C. Greubel, Feb 06 2022

Crossrefs

Cf. A007318 (m=0), A157275 (m=1), A157277 (m=2), this sequence (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274.

Sequence in context: A168622 A168293 A157633 * A144441 A157150 A142461

Adjacent sequences: A157275 A157276 A157277 * A157279 A157280 A157281

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 26 2009

Extensions

Edited by G. C. Greubel, Feb 06 2022

Status

approved