A157272 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+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 1, read by rows.

1, 1, 1, 1, 7, 1, 1, 20, 20, 1, 1, 47, 155, 47, 1, 1, 102, 753, 753, 102, 1, 1, 213, 3004, 7109, 3004, 213, 1, 1, 436, 10800, 48727, 48727, 10800, 436, 1, 1, 883, 36517, 280736, 551251, 280736, 36517, 883, 1, 1, 1778, 118795, 1454163, 4879214, 4879214, 1454163, 118795, 1778, 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+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 1.

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

Example

Triangle begins as:
  1;
  1,    1;
  1,    7,      1;
  1,   20,     20,       1;
  1,   47,    155,      47,       1;
  1,  102,    753,     753,     102,       1;
  1,  213,   3004,    7109,    3004,     213,       1;
  1,  436,  10800,   48727,   48727,   10800,     436,      1;
  1,  883,  36517,  280736,  551251,  280736,   36517,    883,    1;
  1, 1778, 118795, 1454163, 4879214, 4879214, 1454163, 118795, 1778, 1;

Mathematica

f[n_, k_]:= If[k<=Floor[n/2], 2*k+1, 2*(n-k)+1];
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 *)

Prog

(Sage)
def f(n, k): return 2*k+1 if (k <= n//2) else 2*(n-k)+1
@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

Crossrefs

Cf. A007318 (m=0), this sequence (m=1), A157273 (m=2), A157274 (m=3).

Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157275, A157277, A157278.

Sequence in context: A154233 A174033 A119727 * A176200 A046739 A056752

Adjacent sequences: A157269 A157270 A157271 * A157273 A157274 A157275

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 26 2009

Extensions

Edited by G. C. Greubel, Feb 04 2022

Status

approved