A119947 Triangle of numerators in the square of the matrix A[i,j] = 1/i for j <= i, 0 otherwise.

1, 3, 1, 11, 5, 1, 25, 13, 7, 1, 137, 77, 47, 9, 1, 49, 29, 19, 37, 11, 1, 363, 223, 153, 319, 107, 13, 1, 761, 481, 341, 743, 533, 73, 15, 1, 7129, 4609, 3349, 2509, 1879, 275, 191, 17, 1, 7381, 4861, 3601, 2761, 2131, 1627, 1207, 121, 19, 1, 83711, 55991, 42131, 32891, 25961

Offset: 1

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• frac
• tabl

Comments

The triangle of the corresponding denominators is A119948. The rationals appear in lowest terms (while in A027446 they are row-wise on the least common denominator).

The triangle with row number i multiplied with the least common multiple (LCM) of its denominators yields A027446.

First column is A001008. - Tilman Neumann, Oct 01 2008

Column 2 is A064169. - Clark Kimberling, Aug 13 2012

Third diagonal (11, 13, 47, ...) is A188386. - Clark Kimberling, Aug 13 2012

Examples

			The rationals are [1]; [3/4, 1/4]; [11/18, 5/18, 1/9]; [25/48, 13/48, 7/48, 1/16]; ... See the W. Lang link for more.
From _Clark Kimberling_, Aug 13 2012: (Start)
As a triangle given by f(n,m) = Sum_{h=m..n} 1/h, the first six rows are:
    1
    3    1
   11    5    1
   25   13    7    1
  137   77   47    9    1
   49   29   19   37   11    1
  363  223  153  319  107   13    1
(End)
		

Links

• Wolfdieter Lang, First ten rows and rationals.

Crossrefs

Cf. A002024: i appears i times (denominators in row i of the matrix A).

Row sums give A119949. Row sums of the triangle of rationals always give 1.

For the cube of this matrix see the rational triangle A119935/A119932 and A027447; see A027448 for the fourth power.

Cf. A001008, A027446, A064169, A119948, A188386.

Programs

Mathematica

f[n_, m_] := Numerator[Sum[1/k, {k, m, n}]]
Flatten[Table[f[n, m], {n, 1, 10}, {m, 1, n}]]
TableForm[Table[f[n, m], {n, 1, 10}, {m, 1, n}]] (* Clark Kimberling, Aug 13 2012 *)

PARI

A119947_upto(n)={my(M=matrix(n,n,i,j,(j<=i)/i)^2);vector(n,r,apply(numerator,M[r,1..r]))} \\ M. F. Hasler, Nov 05 2019

Formula

a(i,j) = numerator(r(i,j)) with r(i,j):=(A^2)[i,j], where the matrix A has elements a[i,j] = 1/i if j<=i, 0 if j>i, (lower triangular).

Extensions

Edited by M. F. Hasler, Nov 05 2019