A157210 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 = 1, read by rows.

1, 1, 1, 1, 3, 1, 1, 8, 8, 1, 1, 19, 42, 19, 1, 1, 42, 186, 186, 42, 1, 1, 89, 730, 1362, 730, 89, 1, 1, 184, 2640, 8540, 8540, 2640, 184, 1, 1, 375, 9030, 47810, 79952, 47810, 9030, 375, 1, 1, 758, 29722, 246530, 652460, 652460, 246530, 29722, 758, 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 = 1.

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

T(n, 1, 1) = A079583(n-1). - G. C. Greubel, Jan 10 2022

Example

Triangle begins as:
  1;
  1,    1;
  1,    3,     1;
  1,    8,     8,       1;
  1,   19,    42,      19,       1;
  1,   42,   186,     186,      42,       1;
  1,   89,   730,    1362,     730,      89,       1;
  1,  184,  2640,    8540,    8540,    2640,     184,       1;
  1,  375,  9030,   47810,   79952,   47810,    9030,     375,     1;
  1,  758, 29722,  246530,  652460,  652460,  246530,   29722,   758,    1;
  1, 1525, 95238, 1196806, 4796770, 7429760, 4796770, 1196806, 95238, 1525, 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, 1], {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):  # A157210
    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..20)]) # G. C. Greubel, Jan 10 2022

Crossrefs

Cf. A007318 (m=0), this sequence (m=1), A157211 (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. A079583.

Sequence in context: A097712 A238688 A174117 * A359573 A359575 A034801

Adjacent sequences: A157207 A157208 A157209 * A157211 A157212 A157213

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 25 2009

Extensions

Edited by G. C. Greubel, Jan 10 2022

Status

approved