A157154 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, 5, 1, 1, 21, 21, 1, 1, 85, 234, 85, 1, 1, 341, 2110, 2110, 341, 1, 1, 1365, 17163, 35882, 17163, 1365, 1, 1, 5461, 131751, 505979, 505979, 131751, 5461, 1, 1, 21845, 976876, 6395471, 11433118, 6395471, 976876, 21845, 1, 1, 87381, 7089360, 75400800, 220599330, 220599330, 75400800, 7089360, 87381, 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 = 1.

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

T(n, 1, 3) = A002450(n). - G. C. Greubel, Jan 10 2022

Example

Triangle begins as:
  1;
  1,     1;
  1,     5,       1;
  1,    21,      21,        1;
  1,    85,     234,       85,         1;
  1,   341,    2110,     2110,       341,         1;
  1,  1365,   17163,    35882,     17163,      1365,        1;
  1,  5461,  131751,   505979,    505979,    131751,     5461,       1;
  1, 21845,  976876,  6395471,  11433118,   6395471,   976876,   21845,     1;
  1, 87381, 7089360, 75400800, 220599330, 220599330, 75400800, 7089360, 87381, 1;

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 10 2022 *)

Prog

(Sage)
@CachedFunction
def T(n, k, m):  # A157154
    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 10 2022

Crossrefs

Cf. A007318 (m=0), A157152 (m=1), A157153 (m=2), this sequence (m=3), A157155 (m=4), A157156 (m=5).

Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

Cf. A002450.

Sequence in context: A036969 A080249 A333143 * A022168 A359993 A157212

Adjacent sequences: A157151 A157152 A157153 * A157155 A157156 A157157

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved