A157149 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 = 3, read by rows.

1, 1, 1, 1, 11, 1, 1, 57, 57, 1, 1, 247, 930, 247, 1, 1, 1013, 10006, 10006, 1013, 1, 1, 4083, 89139, 225230, 89139, 4083, 1, 1, 16369, 719691, 3771323, 3771323, 719691, 16369, 1, 1, 65519, 5495836, 53239541, 108865438, 53239541, 5495836, 65519, 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 = 3.

T(n, n-k, 3) = T(n, k, 3).

T(n, 1, 3) = A289255(n). - G. C. Greubel, Jan 09 2022

Example

Triangle begins as:
  1;
  1,     1;
  1,    11,       1;
  1,    57,      57,        1;
  1,   247,     930,      247,         1;
  1,  1013,   10006,    10006,      1013,        1;
  1,  4083,   89139,   225230,     89139,     4083,       1;
  1, 16369,  719691,  3771323,   3771323,   719691,   16369,     1;
  1, 65519, 5495836, 53239541, 108865438, 53239541, 5495836, 65519, 1;

Maple

A157149 := proc(n, k)
    option remember;
    if k < 0 or k> n then 0;
    elif k = 0 or k = n then 1;
    else (3*(n-k)+1)*procname(n-1, k-1) + (3*k+1)*procname(n-1, k) + 3*k*(n-k)*procname(n-2, k-1);
    end if;
end proc:
seq(seq(A157149(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, 3], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 09 2022 *)

Prog

(Sage)
@CachedFunction
def T(n, k, m):  # A157149
    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, 3) 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), this sequence (m=3), A157150 (m=4), A157151 (m=5).

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

Cf. A289255.

Sequence in context: A144440 A157209 A014469 * A342889 A166979 A022174

Adjacent sequences: A157146 A157147 A157148 * A157150 A157151 A157152

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Jan 09 2022

Status

approved