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

Author

Roger L. Bagula, Feb 26 2009

Keywords

• nonn
• tabl

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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.

Programs

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 *)

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

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

Extensions

Edited by G. C. Greubel, Feb 04 2022