A144445 Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*T(n-2, k-1) with T(n, 1) = T(n, n) = 1.

1, 1, 1, 1, 23, 1, 1, 206, 206, 1, 1, 1677, 6341, 1677, 1, 1, 13452, 133451, 133451, 13452, 1, 1, 107659, 2403612, 5916231, 2403612, 107659, 1, 1, 861322, 40024068, 200795987, 200795987, 40024068, 861322, 1, 1, 6890633, 638151479, 5875203446, 11687580863, 5875203446, 638151479, 6890633, 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,     23,        1;
  1,    206,      206,         1;
  1,   1677,     6341,      1677,         1;
  1,  13452,   133451,    133451,     13452,        1;
  1, 107659,  2403612,   5916231,   2403612,   107659,      1;
  1, 861322, 40024068, 200795987, 200795987, 40024068, 861322, 1;
		

Links

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

Crossrefs

Cf. A144431, A144432, A144435, A144436, A144438, A144439, A144440, A144441, A144442, A144443, A144444.

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,7,7], {n,15}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 04 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 A144445(n,k): return T(n,k,7,7)
flatten([[A144445(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 04 2022

Formula

T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*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) = (7*n-12)*s(n-1) + 7*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) = (1/7)*(23*8^(n-2) - (7*n+2)).

T(n, 3) = (1/98)*(49*n^2 - 21*n - 59 - 46*(56*n-33)*8^(n-3) + 5989*15^(n-3)). (End)