A157275 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, 5604, 2413, 182, 1, 1, 373, 8679, 38117, 38117, 8679, 373, 1, 1, 756, 29376, 219020, 426002, 219020, 29376, 756, 1, 1, 1523, 95668, 1133786, 3749066, 3749066, 1133786, 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), for n >= 1. - G. C. Greubel, Feb 05 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,    5604,    2413,     182,       1;
  1,  373,  8679,   38117,   38117,    8679,     373,     1;
  1,  756, 29376,  219020,  426002,  219020,   29376,   756,    1;
  1, 1523, 95668, 1133786, 3749066, 3749066, 1133786, 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 05 2022 *)

Prog

(Sage)
def f(n, k): return 2*k if (k <= n//2) else 2*(n-k)
@CachedFunction
def T(n, k, m):  # A157275
    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 05 2022

Crossrefs

Cf. A007318 (m=0), this sequence (m=1), A157277 (m=2), A157278 (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274.

Cf. A101945.

Sequence in context: A119726 A103999 A154985 * A157268 A146959 A157632

Adjacent sequences: A157272 A157273 A157274 * A157276 A157277 A157278

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 26 2009

Extensions

Edited by G. C. Greubel, Feb 05 2022

Status

approved