cp's OEIS Frontend

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

%I A119947 #30 Nov 06 2019 12:41:23
%S A119947 1,3,1,11,5,1,25,13,7,1,137,77,47,9,1,49,29,19,37,11,1,363,223,153,
%T A119947 319,107,13,1,761,481,341,743,533,73,15,1,7129,4609,3349,2509,1879,
%U A119947 275,191,17,1,7381,4861,3601,2761,2131,1627,1207,121,19,1,83711,55991,42131,32891,25961
%N A119947 Triangle of numerators in the square of the matrix A[i,j] = 1/i for j <= i, 0 otherwise.
%C A119947 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).
%C A119947 The triangle with row number i multiplied with the least common multiple (LCM) of its denominators yields A027446.
%C A119947 First column is A001008. - _Tilman Neumann_, Oct 01 2008
%C A119947 Column 2 is A064169. - _Clark Kimberling_, Aug 13 2012
%C A119947 Third diagonal (11, 13, 47, ...) is A188386. - _Clark Kimberling_, Aug 13 2012
%H A119947 Wolfdieter Lang, <a href="/A119947/a119947.txt">First ten rows and rationals.</a>
%F A119947 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).
%e A119947 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.
%e A119947 From _Clark Kimberling_, Aug 13 2012: (Start)
%e A119947 As a triangle given by f(n,m) = Sum_{h=m..n} 1/h, the first six rows are:
%e A119947     1
%e A119947     3    1
%e A119947    11    5    1
%e A119947    25   13    7    1
%e A119947   137   77   47    9    1
%e A119947    49   29   19   37   11    1
%e A119947   363  223  153  319  107   13    1
%e A119947 (End)
%t A119947 f[n_, m_] := Numerator[Sum[1/k, {k, m, n}]]
%t A119947 Flatten[Table[f[n, m], {n, 1, 10}, {m, 1, n}]]
%t A119947 TableForm[Table[f[n, m], {n, 1, 10}, {m, 1, n}]] (* _Clark Kimberling_, Aug 13 2012 *)
%o A119947 (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
%Y A119947 Cf. A002024: i appears i times (denominators in row i of the matrix A).
%Y A119947 Row sums give A119949. Row sums of the triangle of rationals always give 1.
%Y A119947 For the cube of this matrix see the rational triangle A119935/A119932 and A027447; see A027448 for the fourth power.
%Y A119947 Cf. A001008, A027446, A064169, A119948, A188386.
%K A119947 nonn,easy,frac,tabl
%O A119947 1,2
%A A119947 _Wolfdieter Lang_, Jul 20 2006
%E A119947 Edited by _M. F. Hasler_, Nov 05 2019