A144441 Triangle T(n,k) read by rows: T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

1, 1, 1, 1, 14, 1, 1, 83, 83, 1, 1, 432, 1550, 432, 1, 1, 2181, 19898, 19898, 2181, 1, 1, 10930, 217887, 523548, 217887, 10930, 1, 1, 54679, 2199237, 10589795, 10589795, 2199237, 54679, 1, 1, 273428, 21203828, 184722860, 362147222, 184722860, 21203828, 273428, 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,     14,        1;
  1,     83,       83,         1;
  1,    432,     1550,       432,         1;
  1,   2181,    19898,     19898,      2181,         1;
  1,  10930,   217887,    523548,    217887,     10930,        1;
  1,  54679,  2199237,  10589795,  10589795,   2199237,    54679,      1;
  1, 273428, 21203828, 184722860, 362147222, 184722860, 21203828, 273428, 1;
		

Links

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

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144442, A144443, A144444, A144445.

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,4,4], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 03 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 A144441(n,k): return T(n,k,4,4)
flatten([[A144441(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 03 2022

Formula

T(n, k) = (4*n-4*k+1)*T(n-1, k-1) + (4*k-3)*T(n-1, k) + 4*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*(2*n-3)*s(n-1) + 4*s(n-2) with s(1) = 1, s(2) = 2.

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

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

T(n, 2) = (1/2)*(7*5^(n-2) - (2*n+1)).

T(n, 3) = (1/8)*(4*n^2 - 5 - 14*(10*n-3)*5^(n-3) + 355*9^(n-3)). (End)