A157208 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, 8, 1, 1, 31, 31, 1, 1, 102, 342, 102, 1, 1, 317, 2548, 2548, 317, 1, 1, 964, 16001, 37724, 16001, 964, 1, 1, 2907, 91877, 423365, 423365, 91877, 2907, 1, 1, 8738, 501032, 4070208, 7922362, 4070208, 501032, 8738, 1, 1, 26233, 2647858, 35556134, 119460466, 119460466, 35556134, 2647858, 26233, 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).

Example

Triangle begins as:
  1;
  1,    1;
  1,    8,      1;
  1,   31,     31,       1;
  1,  102,    342,     102,       1;
  1,  317,   2548,    2548,     317,       1;
  1,  964,  16001,   37724,   16001,     964,      1;
  1, 2907,  91877,  423365,  423365,   91877,   2907,    1;
  1, 8738, 501032, 4070208, 7922362, 4070208, 501032, 8738, 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, 10}, {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):  # A157208
    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..20)]) # G. C. Greubel, Jan 10 2022

Crossrefs

Cf. A007318 (m=0), A157207 (m=1), this sequence (m=2), A157209 (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.

Sequence in context: A142470 A168523 A144439 * A178347 A141686 A185412

Adjacent sequences: A157205 A157206 A157207 * A157209 A157210 A157211

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 25 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved