OFFSET
1,2
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
LINKS
Wolfdieter Lang, First ten rows and rationals.
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).
EXAMPLE
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)
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 *)
PROG
(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
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Jul 20 2006
EXTENSIONS
Edited by M. F. Hasler, Nov 05 2019
STATUS
approved