A157273 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 = 2, read by rows.

1, 1, 1, 1, 12, 1, 1, 47, 47, 1, 1, 154, 590, 154, 1, 1, 477, 4498, 4498, 477, 1, 1, 1448, 28323, 71232, 28323, 1448, 1, 1, 4363, 162313, 816503, 816503, 162313, 4363, 1, 1, 13110, 882764, 7897486, 15979230, 7897486, 882764, 13110, 1, 1, 39353, 4654100, 69030716, 245382470, 245382470, 69030716, 4654100, 39353, 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 = 2.

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

Example

Triangle begins as:
  1;
  1,     1;
  1,    12,       1;
  1,    47,      47,        1;
  1,   154,     590,      154,         1;
  1,   477,    4498,     4498,       477,         1;
  1,  1448,   28323,    71232,     28323,      1448,        1;
  1,  4363,  162313,   816503,    816503,    162313,     4363,       1;
  1, 13110,  882764,  7897486,  15979230,   7897486,   882764,   13110,     1;
  1, 39353, 4654100, 69030716, 245382470, 245382470, 69030716, 4654100, 39353, 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, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 05 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, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 05 2022

Crossrefs

Cf. A007318 (m=0), A157272 (m=1), this sequence (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: A174039 A174148 A155495 * A350729 A168518 A174667

Adjacent sequences: A157270 A157271 A157272 * A157274 A157275 A157276

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 26 2009

Extensions

Edited by G. C. Greubel, Feb 05 2022

Status

approved