A156286 Triangle T(n, k) = (1/k^n)*Product_{j=1..n} ( (k-1)*(k+1)^j + 1 ), read by rows.

1, 1, 10, 1, 140, 1419, 1, 5740, 242649, 3350536, 1, 700280, 165729267, 7853656384, 161827775045, 1, 255602200, 452606628177, 92036999164096, 6040221703554625, 193317016162131576, 1, 279628806800, 4943822199577371, 5392815929021041024, 1352701610289354714125, 132670761753844630766736, 6731905265314349384346775

Offset: 1

Author

Roger L. Bagula, Feb 07 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  1,     10;
  1,    140,      1419;
  1,   5740,    242649,    3350536;
  1, 700280, 165729267, 7853656384, 161827775045;
		

Links

• G. C. Greubel, Rows n = 1..25 of the triangle, flattened

Crossrefs

Cf. A156173.

Programs

Magma

A156286:= func< n,k | (&*[(k-1)*(k+1)^j + 1: j in [1..n]])/k^n >;
[A156286(n,k): k in [1..n], n in [1..10]]; // G. C. Greubel, Jan 02 2022

Mathematica

T[n_, k_]:= (1/k^n)*Product[(k-1)*(k+1)^j +1, {j,n}];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by G. C. Greubel, Jan 02 2022 *)

Sage

def A156286(n,k): return (1/k^n)*product( (k-1)*(k+1)^j +1 for j in (1..n) )
flatten([[A156286(n,k) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, Jan 02 2022

Formula

T(n, k) = Product_{j=1..n} ( (k+1)^j - Sum_{i=0..k-1} (k+1)^i ).

T(n, k) = (1/k^n)*Product_{j=1..n} ( (k-1)*(k+1)^j + 1 ). - G. C. Greubel, Jan 02 2022

Extensions

Edited by G. C. Greubel, Jan 02 2022