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

Author

Roger L. Bagula, Feb 25 2009

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

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.

Programs

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 *)

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

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).

Extensions

Edited by G. C. Greubel, Jan 10 2022