A142461 Triangle read by rows: T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), where m = 6.

1, 1, 1, 1, 14, 1, 1, 111, 111, 1, 1, 796, 2886, 796, 1, 1, 5597, 52642, 52642, 5597, 1, 1, 39210, 824271, 2000396, 824271, 39210, 1, 1, 274507, 11931033, 58614299, 58614299, 11931033, 274507, 1, 1, 1921592, 165260188, 1483533704, 2930714950, 1483533704, 165260188, 1921592, 1

Offset: 1

Author

Roger L. Bagula, Sep 19 2008

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     14,        1;
  1,    111,      111,        1;
  1,    796,     2886,      796,        1;
  1,   5597,    52642,    52642,     5597,        1;
  1,  39210,   824271,  2000396,   824271,    39210,      1;
  1, 274507, 11931033, 58614299, 58614299, 11931033, 274507, 1;
		

Links

• G. C. Greubel, Rows n = 1..50 of the triangle, flattened
• G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_6(n,k).

Crossrefs

For m = ...,-2,-1,0,1,2,3,4,5,6,7, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142460, ...

Cf. A047657 (row sums).

Programs

Mathematica

T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k -m+1)*T[n-1, k, m]];
A142461[n_, k_]:= T[n, k, 6];
Table[A142461[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 17 2022 *)

Sage

@CachedFunction
def T(n,k,m):
    if (k==1 or k==n): return 1
    else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
def A142461(n,k): return T(n,k,6)
flatten([[ A142461(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 2022

Formula

T(n,k,m) = (m*n - m*k + 1)*T(n-1, k-1, m) + (m*k - (m-1))*T(n-1, k, m), with T(n, 1, m) = T(n, n, m) = 1, and m = 6.

Sum_{k=1..n} T(n, k, 6) = A047657(n-1).

Extensions

Edited by N. J. A. Sloane, May 08 2013