A157156 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 = 5, read by rows.

1, 1, 1, 1, 7, 1, 1, 43, 43, 1, 1, 259, 806, 259, 1, 1, 1555, 11720, 11720, 1555, 1, 1, 9331, 151215, 338770, 151215, 9331, 1, 1, 55987, 1828221, 7892635, 7892635, 1828221, 55987, 1, 1, 335923, 21286168, 162474781, 304389070, 162474781, 21286168, 335923, 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 = 5.

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

T(n, 1, 5) = A003464(n). - G. C. Greubel, Jan 10 2022

Example

Triangle begins as:
  1;
  1,      1;
  1,      7,        1;
  1,     43,       43,         1;
  1,    259,      806,       259,         1;
  1,   1555,    11720,     11720,      1555,         1;
  1,   9331,   151215,    338770,    151215,      9331,        1;
  1,  55987,  1828221,   7892635,   7892635,   1828221,    55987,      1;
  1, 335923, 21286168, 162474781, 304389070, 162474781, 21286168, 335923, 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, 5], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *)

Prog

(Sage)
@CachedFunction
def T(n, k, m):  # A157156
    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, 5) 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), A157154 (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. A003464.

Sequence in context: A156916 A173584 A166973 * A022170 A178658 A156602

Adjacent sequences: A157153 A157154 A157155 * A157157 A157158 A157159

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 24 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved