A157211 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) = k if k <= floor(n/2) otherwise n-k, and m = 2, read by rows.

1, 1, 1, 1, 4, 1, 1, 15, 15, 1, 1, 50, 134, 50, 1, 1, 157, 960, 960, 157, 1, 1, 480, 6013, 12636, 6013, 480, 1, 1, 1451, 34717, 136809, 136809, 34717, 1451, 1, 1, 4366, 190528, 1303472, 2361474, 1303472, 190528, 4366, 1, 1, 13113, 1012326, 11392866, 34496986, 34496986, 11392866, 1012326, 13113, 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) = k if k <= floor(n/2) otherwise n-k, and m = 2.

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

T(n, 1, 2) = A132308(n-1). - G. C. Greubel, Jan 10 2022

Example

Triangle begins as:
  1;
  1,     1;
  1,     4,       1;
  1,    15,      15,        1;
  1,    50,     134,       50,        1;
  1,   157,     960,      960,      157,        1;
  1,   480,    6013,    12636,     6013,      480,        1;
  1,  1451,   34717,   136809,   136809,    34717,     1451,       1;
  1,  4366,  190528,  1303472,  2361474,  1303472,   190528,    4366,     1;
  1, 13113, 1012326, 11392866, 34496986, 34496986, 11392866, 1012326, 13113, 1;

Mathematica

f[n_, k_]:= If[k<=Floor[n/2], k, 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, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 10 2022 *)

Prog

(Sage)
def f(n, k): return k if (k <= n//2) else n-k
@CachedFunction
def T(n, k, m):  # A157211
    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, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 10 2022

Crossrefs

Cf. A007318 (m=0), A157210 (m=1), this sequence (m=2), A157212 (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157268, A157272, A157273, A157274, A157275.

Cf. A132308.

Sequence in context: A271705 A320280 A343804 * A176428 A116469 A156599

Adjacent sequences: A157208 A157209 A157210 * A157212 A157213 A157214

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 25 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved