A157212 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, 5, 1, 1, 24, 24, 1, 1, 103, 306, 103, 1, 1, 422, 3028, 3028, 422, 1, 1, 1701, 26064, 57806, 26064, 1701, 1, 1, 6820, 207132, 889640, 889640, 207132, 6820, 1, 1, 27299, 1569298, 11975936, 22436968, 11975936, 1569298, 27299, 1, 1, 109218, 11544744, 147711834, 472619880, 472619880, 147711834, 11544744, 109218, 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,     5,       1;
  1,    24,      24,        1;
  1,   103,     306,      103,        1;
  1,   422,    3028,     3028,      422,        1;
  1,  1701,   26064,    57806,    26064,     1701,       1;
  1,  6820,  207132,   889640,   889640,   207132,    6820,     1;
  1, 27299, 1569298, 11975936, 22436968, 11975936, 1569298, 27299, 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, 12}, {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):  # A157212
    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, Jan 10 2022

Crossrefs

Cf. A007318 (m=0), A157210 (m=1), A157211 (m=2), this sequence (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157268, A157272, A157273, A157274, A157275.

Sequence in context: A157154 A022168 A359993 * A156600 A329118 A152572

Adjacent sequences: A157209 A157210 A157211 * A157213 A157214 A157215

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 25 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved