A119932 -id:A119932 - cp's OEIS frontend

A119933 Row sums of triangle A119932 (denominator triangle for certain rationals).

1, 16, 243, 1792, 56125, 92016, 5260591, 28475392, 290387673, 445906000, 59350089931, 64786127616, 11861260199893, 4143106927792, 3092227583175, 22046047952896, 6652851946041649, 12011298089015952, 4425675600220359715

Offset: 1

Wolfdieter Lang, Jul 20 2006

a(i)=sum( A119932(i,j),j=1..i),i>=1.

A119936 Least common multiple (LCM) of denominators of the rows of the triangle of rationals A119935/A119932.

Original entry on oeis.org

1, 8, 108, 576, 18000, 21600, 1234800, 5644800, 57153600, 63504000, 8452382400, 9220780800, 1688171284800, 1818030614400, 1947889944000, 8310997094400, 2551995545299200, 2702112930316800, 1029655143835718400

Offset: 1

Wolfdieter Lang, Jul 20 2006

The triangle of rationals is the matrix cube of the matrix with elements a(i,j) = 1/i if j <= i, 0 if j > i.

Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012.

Distinct from A246498.

A027447(i,j)= a(i)* A119935(i,j)/A119932(i,j) .
a(n) = lcm_{m=1..n} seq(A119932(n,m)), n >= 1.
a(n)/n^3 = A027451(n) = A002944(n)^2 (the second equation is a conjecture).
a(n)/n^3 = (A099946(n)*(n-1))^2, n >= 2 (from the conjecture).

A119935 Triangle of numerators of the cube of a certain lower triangular matrix.

Original entry on oeis.org

1, 7, 1, 85, 19, 1, 415, 115, 37, 1, 12019, 3799, 1489, 61, 1, 13489, 4669, 2059, 919, 91, 1, 726301, 268921, 128431, 64171, 7669, 127, 1, 3144919, 1227199, 621139, 334699, 178669, 3565, 169, 1, 30300391

Offset: 1

Wolfdieter Lang, Jul 20 2006

The triangle of the corresponding denominators is A119932.

This triangle of numerators is related to (and was derived from) A027447. There the least common multiple (lcm) of the denominators of each row i of the triangle of rationals r(i,j) has been multiplied in order to obtain an integer triangle.

Wolfdieter Lang, First ten rows and rationals.

a(i, j)=1/A002024(i, j), i>=1, j<=i.
Row sums give A119934. Row sums of the triangle of rationals are identical 1.
Cf. A027447.

A119935 := proc(n::integer,k::integer)
    m := Matrix(n,n) ;
    for i from 1 to n do
    for j from 1 to i do
        m[i,j] := 1/i ;
    end do:
    end do:
    m3 := LinearAlgebra[MatrixPower](m,3) ;
    m3[n,k] ;
    numer(%) ;
end proc: # R. J. Mathar, Nov 05 2019

{d↑⍨¯1+(d←⍕⍵)⍳'r'}¨(c≠0)/c←,b+.×b+.×b←a∘.{⍺÷⍨⍺≥⍵}a←⍳20x ⍝ Michael Turniansky, Jan 11 2021

a(i,j) = numerator(r(i,j)) with r(i,j):=(A^3)[i,j], where the lower triangular matrix A has elements a[i,j] = 1/i if j<=i, 0 if j>i.

Offset corrected by R. J. Mathar, Nov 05 2019

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

Original entry on oeis.org

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

Wolfdieter Lang, Jul 20 2006

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

			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)

Wolfdieter Lang, First ten rows and rationals.

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.

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 *)

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

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).

Edited by M. F. Hasler, Nov 05 2019

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A119933 Row sums of triangle A119932 (denominator triangle for certain rationals).

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A119936 Least common multiple (LCM) of denominators of the rows of the triangle of rationals A119935/A119932.

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A119935 Triangle of numerators of the cube of a certain lower triangular matrix.

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

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