A157274 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 = 3, read by rows.

1, 1, 1, 1, 17, 1, 1, 84, 84, 1, 1, 355, 1431, 355, 1, 1, 1442, 14827, 14827, 1442, 1, 1, 5793, 127860, 326591, 127860, 5793, 1, 1, 23200, 1009338, 5239457, 5239457, 1009338, 23200, 1, 1, 92831, 7593061, 71229038, 145043839, 71229038, 7593061, 92831, 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 = 3.

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

Example

Triangle begins as:
  1;
  1,     1;
  1,    17,       1;
  1,    84,      84,        1;
  1,   355,    1431,      355,         1;
  1,  1442,   14827,    14827,      1442,        1;
  1,  5793,  127860,   326591,    127860,     5793,       1;
  1, 23200, 1009338,  5239457,   5239457,  1009338,   23200,     1;
  1, 92831, 7593061, 71229038, 145043839, 71229038, 7593061, 92831, 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, 3], {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, 3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 05 2022

Crossrefs

Cf. A007318 (m=0), A157272 (m=1), A157273 (m=2), this sequence (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: A176203 A103637 A229956 * A218115 A144442 A157151

Adjacent sequences: A157271 A157272 A157273 * A157275 A157276 A157277

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 26 2009

Extensions

Edited by G. C. Greubel, Feb 05 2022

Status

approved