A385850 Triangle read by rows: T(n,k) = denominator((Sum_{i=1..k} (n-i+1)^2)/(Sum_{i=1..k} i^2)), with 1 <= k <= n.

1, 1, 1, 1, 5, 1, 1, 1, 14, 1, 1, 5, 7, 5, 1, 1, 5, 2, 15, 11, 1, 1, 1, 7, 5, 11, 91, 1, 1, 5, 14, 5, 11, 91, 20, 1, 1, 1, 7, 3, 11, 91, 1, 51, 1, 1, 5, 2, 5, 1, 91, 20, 51, 95, 1, 1, 5, 7, 5, 11, 91, 5, 17, 95, 77, 1, 1, 1, 14, 15, 11, 7, 4, 51, 95, 77, 46, 1

Offset

1,5

Links

Stefano Spezia, Table of n, a(n) for n = 1..11325 (first 150 rows of the triangle, flattened)

Formula

T(n,k) = denominator((1 - 3*k + 2*k^2 + 6*n - 6*k*n + 6*n^2)/(1 + 3*k + 2*k^2)).

Example

Triangle of the fractions begins as:
   1/1;
   4/1,  1/1;
   9/1, 13/5,    1/1;
  16/1,  5/1,  29/14,    1/1;
  25/1, 41/5,   25/7,    9/5,    1/1;
  36/1, 61/5,   11/2,  43/15,  18/11,     1/1;
  49/1, 17/1,   55/7,   21/5,  27/11,  139/91,  1/1;
  ...
A385849(4,3)/T(4,3) = (4^2 + 3^2 + 2^2)/(1^2 + 2^2 + 3^2) = 29/14.

Mathematica

T[n_, k_]:=Denominator[(1-3k+2k^2+6n-6k*n+6n^2)/(1+3k+2k^2)]; Table[T[n, k], {n, 12}, {k, n}]//Flatten

Crossrefs

Cf. A000012 (1st column or diagonal), A385849 (numerators).

Sequence in context: A183097 A353958 A285486 * A230368 A256690 A181985

Adjacent sequences: A385847 A385848 A385849 * A385851 A385852 A385853

Keyword

nonn,easy,frac,look,tabl

Author

Stefano Spezia, Jul 10 2025

Status

approved