A144439 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 = 2, and j = 2, read by rows.

1, 1, 1, 1, 8, 1, 1, 31, 31, 1, 1, 102, 326, 102, 1, 1, 317, 2406, 2406, 317, 1, 1, 964, 15087, 34336, 15087, 964, 1, 1, 2907, 86673, 380947, 380947, 86673, 2907, 1, 1, 8738, 473084, 3650206, 6925718, 3650206, 473084, 8738, 1, 1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 1

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 = 2, and j = 2.

Sum_{k=0..n} T(n, k) = s(n), where s(n) = 2*(n-1)*s(n-1) + 2*s(n-2), s(1) = 1, s(2) = 2.

From G. C. Greubel, Mar 10 2022: (Start)

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

T(n, 2) = 4*3^(n-2) - (n+1).

T(n, 3) = (1/2)*(71*5^(n-3) - 8*(3*n+1)*3^(n-3) + n^2 + n - 1). (End)

Example

Triangle begins as:
  1;
  1,     1;
  1,     8,       1;
  1,    31,      31,        1;
  1,   102,     326,      102,         1;
  1,   317,    2406,     2406,       317,         1;
  1,   964,   15087,    34336,     15087,       964,        1;
  1,  2907,   86673,   380947,    380947,     86673,     2907,       1;
  1,  8738,  473084,  3650206,   6925718,   3650206,   473084,    8738,     1;
  1, 26233, 2502304, 31874880, 103245622, 103245622, 31874880, 2502304, 26233, 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, 2, 2], {n, 15}, {k, n}]//Flatten (* modified by G. C. Greubel, Mar 10 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 A144439(n, k): return T(n, k, 2, 2)
flatten([[A144439(n, k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 10 2022

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144440, A144441, A144442, A144443, A144444, A144445, A144446, A144447.

Sequence in context: A178122 A142470 A168523 * A157208 A178347 A141686

Adjacent sequences: A144436 A144437 A144438 * A144440 A144441 A144442

Keyword

nonn,tabl

Author

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

Status

approved