A142458 Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).

1, 1, 1, 1, 8, 1, 1, 39, 39, 1, 1, 166, 546, 166, 1, 1, 677, 5482, 5482, 677, 1, 1, 2724, 47175, 109640, 47175, 2724, 1, 1, 10915, 373809, 1709675, 1709675, 373809, 10915, 1, 1, 43682, 2824048, 23077694, 44451550, 23077694, 2824048, 43682, 1

Offset: 1

Author

Roger L. Bagula, Sep 19 2008

Keywords

• nonn
• easy
• tabl

Comments

Consider the triangle 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). For m = ...,-2,-1,0,1,2,3,... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, ... - N. J. A. Sloane, May 08 2013

Examples

			The rows n >= 1 and columns 1 <= k <= n look as follows:
  1;
  1,     1;
  1,     8,       1;
  1,    39,      39,        1;
  1,   166,     546,      166,        1;
  1,   677,    5482,     5482,      677,        1;
  1,  2724,   47175,   109640,    47175,     2724,       1;
  1, 10915,  373809,  1709675,  1709675,   373809,   10915,     1;
  1, 43682, 2824048, 23077694, 44451550, 23077694, 2824048, 43682, 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_3(n,k).

Crossrefs

Cf. A225372 (m=-2), A144431 (m=-1), A007318 (m=0), A008292 (m=1), A060187 (m=2), this sequence (m=3), A142459 (m=4), A142560 (m=5), A142561 (m=6), A142562 (m=7), A167884 (m=8), A257608 (m=9).

Cf. A008544, A142976, A144380, A144381, A144414.

Programs

Maple

A142458 := proc(n,k) if n = k then 1; elif k > n or k < 1 then 0 ;else (3*n-3*k+1)*procname(n-1,k-1)+(3*k-2)*procname(n-1,k) ; end if; end proc:
seq(seq(A142458(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Jun 04 2011

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] ];
Table[T[n, k, 3], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by G. C. Greubel, Mar 14 2022 *)

Sage

def T(n,k,m): # A142458
    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)
flatten([[T(n,k,3) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Mar 14 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 = 3.

Sum_{k=1..n} T(n, k) = A008544(n-1).

From G. C. Greubel, Mar 14 2022: (Start)

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

T(n, 2) = A144414(n-1).

T(n, 3) = A142976(n-2).

T(n, 4) = A144380(n-3).

T(n, 5) = A144381(n-4). (End)

Extensions

Edited by the Associate Editors of the OEIS, Aug 28 2009