A157150 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) + m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4, read by rows.

1, 1, 1, 1, 14, 1, 1, 87, 87, 1, 1, 460, 1790, 460, 1, 1, 2333, 24178, 24178, 2333, 1, 1, 11706, 271983, 693068, 271983, 11706, 1, 1, 58579, 2786993, 14794139, 14794139, 2786993, 58579, 1, 1, 292952, 27109300, 267169640, 547357078, 267169640, 27109300, 292952, 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*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 4.

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

Example

Triangle begins as:
  1;
  1,      1;
  1,     14,        1;
  1,     87,       87,         1;
  1,    460,     1790,       460,         1;
  1,   2333,    24178,     24178,      2333,         1;
  1,  11706,   271983,    693068,    271983,     11706,        1;
  1,  58579,  2786993,  14794139,  14794139,   2786993,    58579,      1;
  1, 292952, 27109300, 267169640, 547357078, 267169640, 27109300, 292952, 1;

Maple

A157150:= proc(n, k);
    if k<0 or n<k then 0;
    elif k=0 or k=n then 1;
    else (4*n-4*k+1)*procname(n-1, k-1) + (4*k+1)*procname(n-1, k) + 4*k*(n-k)*procname(n-2, k-1);
    end if; end proc;
seq(seq(A157150(n, k), k=0..n), n=0..10); # R. J. Mathar, Feb 06 2015

Mathematica

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*k*(n-k)*T[n-2, k-1, m]];
Table[T[n, k, 4], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

Prog

(Sage)
@CachedFunction
def T(n, k, m):  # A157150
    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*k*(n-k)*T(n-2, k-1, m)
flatten([[T(n, k, 4) for k in (0..n)] for n in (0..20)]) # G. C. Greubel, Jan 09 2022

Crossrefs

Cf. A007318 (m=0), A157147 (m=1), A157148 (m=2), A157149 (m=3), this sequence (m=4), A157151 (m=5).

Cf. A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

Sequence in context: A157633 A157278 A144441 * A142461 A174720 A060628

Adjacent sequences: A157147 A157148 A157149 * A157151 A157152 A157153

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Jan 09 2022

Status

approved