A142462 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 = 7.

1, 1, 1, 1, 16, 1, 1, 143, 143, 1, 1, 1166, 4290, 1166, 1, 1, 9357, 90002, 90002, 9357, 1, 1, 74892, 1621383, 3960088, 1621383, 74892, 1, 1, 599179, 27016857, 134142043, 134142043, 27016857, 599179, 1, 1, 4793482, 431017552, 3923731798, 7780238494, 3923731798, 431017552, 4793482, 1

Offset: 1

Author

Roger L. Bagula, Sep 19 2008

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     16,        1;
  1,    143,      143,         1;
  1,   1166,     4290,      1166,         1;
  1,   9357,    90002,     90002,      9357,        1;
  1,  74892,  1621383,   3960088,   1621383,    74892,      1;
  1, 599179, 27016857, 134142043, 134142043, 27016857, 599179, 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_7(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, A142461, A142462, ...

Cf. A084947 (row sums).

Programs

Magma

function T(n,k,m)
  if k eq 1 or k eq n then return 1;
  else return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m);
  end if; return T;
end function;
A142462:= func< n,k | T(n,k,7) >;
[A142462(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 17 2022

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]];
A142462[n_, k_]:= T[n,k,7];
Table[A142462[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 A142462(n,k): return T(n,k,7)
flatten([[ A142462(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Mar 17 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 = 7.

Sum_{k=1..n} T(n, k) = A084947(n-1).

Extensions

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