A144443 Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

1, 1, 1, 1, 20, 1, 1, 159, 159, 1, 1, 1138, 4254, 1138, 1, 1, 7997, 77878, 77878, 7997, 1, 1, 56016, 1219167, 2984888, 1219167, 56016, 1, 1, 392155, 17633649, 87659315, 87659315, 17633649, 392155, 1, 1, 2745134, 244083268, 2219485106, 4400875078, 2219485106, 244083268, 2745134, 1

Offset

1,5

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

Sum_{k=1..n} T(n, k) = s(n), where s(n) = 2*(3*n-5)*s(n-1) + 6*s(n-2), s(1) = 1, s(2) = 2.

From G. C. Greubel, Mar 04 2022: (Start)

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

T(n, 2) = (1/3)*(10*7^(n-2) - (3*n+1)).

T(n, 3) = (1/18)*(9*n^2 -3*n -11 - 20*(21*n-11)*7^(n-3) + 997*13^(n-3)). (End)

Example

Triangle begins as:
  1;
  1,      1;
  1,     20,        1;
  1,    159,      159,        1;
  1,   1138,     4254,     1138,        1;
  1,   7997,    77878,    77878,     7997,        1;
  1,  56016,  1219167,  2984888,  1219167,    56016,      1;
  1, 392155, 17633649, 87659315, 87659315, 17633649, 392155, 1;

Mathematica

T[n_, k_, m_, j_]:= T[n, k, m, j]= If[k==1 || k==n, 1, (m*(n-k)+1)*T[n-1, k-1, m, j] + (m*(k-1)+1)*T[n-1, k, m, j] + j*T[n-2, k-1, m, j]];
Table[T[n, k, 6, 6], {n, 15}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 04 2022 *)

Prog

(Sage)
def T(n, k, m, j):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1, k-1, m, j) + (m*(k-1)+1)*T(n-1, k, m, j) + j*T(n-2, k-1, m, j)
def A144443(n, k): return T(n, k, 6, 6)
flatten([[A144443(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 04 2022

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144444, A144445.

Sequence in context: A154652 A155516 A174674 * A257608 A022183 A015146

Adjacent sequences: A144440 A144441 A144442 * A144444 A144445 A144446

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 05 2008

Status

approved