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

Author

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

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

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

Programs

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 *)

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

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)