A142460 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 = 5.

1, 1, 1, 1, 12, 1, 1, 83, 83, 1, 1, 514, 1826, 514, 1, 1, 3105, 28310, 28310, 3105, 1, 1, 18656, 376615, 905920, 376615, 18656, 1, 1, 111967, 4627821, 22403635, 22403635, 4627821, 111967, 1, 1, 671838, 54377008, 478781506, 940952670, 478781506, 54377008, 671838, 1

Offset: 1

Author

Roger L. Bagula, Sep 19 2008

Keywords

• nonn
• tabl
• easy

Comments

One of a family of triangles. For m = ...,-2,-1,0,1,2,3,4,5,... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, A142459, A142560, ...

Examples

			Triangle begins as:
  1;
  1,      1;
  1,     12,        1;
  1,     83,       83,         1;
  1,    514,     1826,       514,         1;
  1,   3105,    28310,     28310,      3105,         1;
  1,  18656,   376615,    905920,    376615,     18656,        1;
  1, 111967,  4627821,  22403635,  22403635,   4627821,   111967,      1;
  1, 671838, 54377008, 478781506, 940952670, 478781506, 54377008, 671838, 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_5(n,k).

Crossrefs

Cf. A225372 (m=-2), A144431 (m=-1), A007318 (m=0), A008292 (m=1), A060187 (m=2), A142458 (m=3), A142459 (m=4), this sequence (m=5), A142561 (m=6), A142562 (m=7), A167884 (m=8), A257608 (m=9).

Cf. A047055 (row sums).

Programs

Maple

A142460 := proc(n, k) if n = k then 1; elif k > n or k < 1 then 0 ; else (5*n-5*k+1)*procname(n-1, k-1)+(5*k-4)*procname(n-1, k) ; end if; end proc:
seq(seq(A142459(n, k), k=1..n), n=1..10) ; # R. J. Mathar, May 11 2013

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, 5], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by G. C. Greubel, Mar 14 2022 *)

Sage

def T(n,k,m): # A142460
    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,5) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Mar 14 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(t,1,m) = T(n,n,m) = 1, and m = 5.

Sum_{k=1..n} T(n, k, 5) = A047055(n-1).

Extensions

Edited by N. J. A. Sloane, May 08 2013, May 11 2013