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

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

Author

Wolfdieter Lang, Jul 20 2006

Keywords

• nonn
• easy
• tabl
• frac

Comments

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.

Links

• Wolfdieter Lang, First ten rows and rationals.

Crossrefs

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.

Programs

Maple

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

NARS2000

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

Formula

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.

Extensions

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