A144436 Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.

1, 1, 1, 1, 8, 1, 1, 23, 23, 1, 1, 54, 170, 54, 1, 1, 117, 818, 818, 117, 1, 1, 244, 3255, 7224, 3255, 244, 1, 1, 499, 11697, 48443, 48443, 11697, 499, 1, 1, 1010, 39560, 276974, 513326, 276974, 39560, 1010, 1, 1, 2033, 128756, 1431604, 4422246

Offset

1,5

Links

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

Formula

T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4.

From G. C. Greubel, Mar 03 2022: (Start)

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

T(n, 2) = 2^(n+1) - (n+5).

T(n, 3) = (1/2)*( n^2 + 9*n + 16 - 2^(n+2)*(n+3) + 142*3^(n-3) ). (End)

Example

Triangle begins as:
  1;
  1,    1;
  1,    8,      1;
  1,   23,     23,       1;
  1,   54,    170,      54,       1;
  1,  117,    818,     818,     117,       1;
  1,  244,   3255,    7224,    3255,     244,       1;
  1,  499,  11697,   48443,   48443,   11697,     499,      1;
  1, 1010,  39560,  276974,  513326,  276974,   39560,   1010,    1;
  1, 2033, 128756, 1431604, 4422246, 4422246, 1431604, 128756, 2033, 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, 1, 4], {n, 15}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 03 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 A144436(n, k): return T(n, k, 1, 4)
flatten([[A144436(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

Crossrefs

Cf. A144431, A144432, A144435.

Sequence in context: A157178 A174303 A176488 * A358628 A167034 A155452

Adjacent sequences: A144433 A144434 A144435 * A144437 A144438 A144439

Keyword

nonn,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 04 2008

Extensions

Edited by G. C. Greubel, Mar 03 2022

Status

approved