A167884 Triangle read by rows: T(n,k) 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 = 8.

1, 1, 1, 1, 18, 1, 1, 179, 179, 1, 1, 1636, 6086, 1636, 1, 1, 14757, 144362, 144362, 14757, 1, 1, 132854, 2941135, 7218100, 2941135, 132854, 1, 1, 1195735, 55446309, 277509955, 277509955, 55446309, 1195735, 1, 1, 10761672, 1001178268, 9211047544, 18315657030, 9211047544, 1001178268, 10761672, 1

Offset: 1

Author

Roger L. Bagula, Nov 14 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,       1;
  1,      18,        1;
  1,     179,      179,         1;
  1,    1636,     6086,      1636,         1;
  1,   14757,   144362,    144362,     14757,        1;
  1,  132854,  2941135,   7218100,   2941135,   132854,       1;
  1, 1195735, 55446309, 277509955, 277509955, 55446309, 1195735, 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_8(n,k)

Crossrefs

For m = ...,-2,-1,0,1,2,3,4,5,6,7,8, ... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142460, A142461, A142462, A167884, ...

Cf. A084948 (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]];
A167884[n_, k_]:= T[n,k,8];
Table[A167884[n, k], {n,12}, {k,n}]//Flatten (* modified by G. C. Greubel, Mar 18 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 A167884(n,k): return T(n,k,8)
flatten([[ A167884(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 18 2022

Formula

T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 8.

Sum_{k=1..n} T(n, k) = A084948(n-1).

Extensions

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