A157268 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) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1, read by rows.

1, 1, 1, 1, 6, 1, 1, 17, 17, 1, 1, 40, 126, 40, 1, 1, 87, 606, 606, 87, 1, 1, 182, 2413, 5856, 2413, 182, 1, 1, 373, 8679, 40337, 40337, 8679, 373, 1, 1, 756, 29376, 232726, 497066, 232726, 29376, 756, 1, 1, 1523, 95668, 1205968, 4527078, 4527078, 1205968, 95668, 1523, 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*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2^k if k <= floor(n/2) otherwise 2^(n-k), and m = 1.

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

T(n, 1, 1) = A101945(n-1), n >= 1. - G. C. Greubel, Feb 04 2022

Example

Triangle begins as:
  1;
  1,    1;
  1,    6,     1;
  1,   17,    17,       1;
  1,   40,   126,      40,       1;
  1,   87,   606,     606,      87,       1;
  1,  182,  2413,    5856,    2413,     182,       1;
  1,  373,  8679,   40337,   40337,    8679,     373,     1;
  1,  756, 29376,  232726,  497066,  232726,   29376,   756,    1;
  1, 1523, 95668, 1205968, 4527078, 4527078, 1205968, 95668, 1523, 1;

Mathematica

f[n_, k_]:= If[k<=Floor[n/2], 2^k, 2^(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, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 04 2022 *)

Prog

(Sage)
def f(n, k): return 2^k if (k <= n//2) else 2^(n-k)
@CachedFunction
def T(n, k, m):  # A157207
    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, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 04 2022

Crossrefs

Cf. A007318 (m=0), this sequence (m=1).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157272, A157273, A157274, A157275, A157277, A157278.

Cf. A101945.

Sequence in context: A103999 A154985 A157275 * A146959 A157632 A328888

Adjacent sequences: A157265 A157266 A157267 * A157269 A157270 A157271

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 26 2009

Extensions

Edited by G. C. Greubel, Feb 04 2022

Status

approved