%I #6 Sep 08 2022 08:45:41

%S 1,1,10,1,140,1419,1,5740,242649,3350536,1,700280,165729267,

%T 7853656384,161827775045,1,255602200,452606628177,92036999164096,

%U 6040221703554625,193317016162131576,1,279628806800,4943822199577371,5392815929021041024,1352701610289354714125,132670761753844630766736,6731905265314349384346775

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

%H G. C. Greubel, <a href="/A156286/b156286.txt">Rows n = 1..25 of the triangle, flattened</a>

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

%F T(n, k) = (1/k^n)*Product_{j=1..n} ( (k-1)*(k+1)^j + 1 ). - _G. C. Greubel_, Jan 02 2022

%e Triangle begins as:

%e 1;

%e 1, 10;

%e 1, 140, 1419;

%e 1, 5740, 242649, 3350536;

%e 1, 700280, 165729267, 7853656384, 161827775045;

%t T[n_, k_]:= (1/k^n)*Product[(k-1)*(k+1)^j +1, {j,n}];

%t Table[T[n, k], {n, 10}, {k, n}]//Flatten (* modified by _G. C. Greubel_, Jan 02 2022 *)

%o (Magma) A156286:= func< n,k | (&*[(k-1)*(k+1)^j + 1: j in [1..n]])/k^n >;

%o [A156286(n,k): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Jan 02 2022

%o (Sage)

%o def A156286(n,k): return (1/k^n)*product( (k-1)*(k+1)^j +1 for j in (1..n) )

%o flatten([[A156286(n,k) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Jan 02 2022

%Y Cf. A156173.

%K nonn,tabl

%O 1,3

%A _Roger L. Bagula_, Feb 07 2009

%E Edited by _G. C. Greubel_, Jan 02 2022