A385849 Triangle read by rows: T(n,k) = numerator((Sum_{i=1..k} (n-i+1)^2)/(Sum_{i=1..k} i^2)), with 1 <= k <= n.

1, 4, 1, 9, 13, 1, 16, 5, 29, 1, 25, 41, 25, 9, 1, 36, 61, 11, 43, 18, 1, 49, 17, 55, 21, 27, 139, 1, 64, 113, 149, 29, 38, 199, 29, 1, 81, 29, 97, 23, 51, 271, 2, 71, 1, 100, 181, 35, 49, 6, 355, 53, 95, 128, 1, 121, 221, 151, 61, 83, 451, 17, 41, 167, 101, 1

Offset

1,2

Links

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

Michael De Vlieger, Scatterplot of a(n), n = 1..11325.

Formula

T(n,k) = numerator((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;
  ...
T(4,3)/A385850(4,3) = (4^2 + 3^2 + 2^2)/(1^2 + 2^2 + 3^2) = 29/14.

Mathematica

T[n_, k_]:=Numerator[(1-3k+2k^2+6n-6k*n+6n^2)/(1+3k+2k^2)]; Table[T[n, k], {n, 11}, {k, n}]//Flatten

Crossrefs

Cf. A000012 (diagonal), A000290 (1st column), A322135, A385850 (denominators).

Sequence in context: A091885 A069606 A344109 * A193580 A244761 A075150

Adjacent sequences: A385846 A385847 A385848 * A385850 A385851 A385852

Keyword

nonn,easy,frac,look,tabl

Author

Stefano Spezia, Jul 10 2025

Status

approved