A229557 -id:A229557 - cp's OEIS frontend

A229556 Array read by antidiagonals. Rows are the numerators of consecutive harmonic transforms starting with a first row 1, 1, 1, ....

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 11, 5, 1, 1, 5, 25, 73, 8, 1, 1, 6, 137, 2221, 749, 13, 1, 1, 7, 49, 353777, 1964654, 12657, 21, 1, 1, 8, 363, 19595573, 786674809783, 14862065179, 343693, 34, 1, 1, 9, 761, 239046803, 17003676861538314284, 13379715149864207035877, 35955580499839

Offset: 1

Franz Vrabec, Sep 26 2013

The "harmonic transform" of a sequence of positive numbers a(i) is the sequence h(n) of the partial sums of their reciprocals: h(n) = Sum_{i=1..n} 1/a(i).

			Table begins
  1, 1,   1,       1, ...
  1, 2,   3,       4, ...
  1, 3,  11,      25, ...
  1, 5,  73,    2221, ...
  1, 8, 749, 1964654, ...
which are the numerators of
  1,   1,       1,              1,                         1, ...
  1,   2,       3,              4,                         5, ...
  1, 3/2,    11/6,          25/12,                    137/60, ...
  1, 5/3,   73/33,       2221/825,             353777/113025, ...
  1, 8/5, 749/365, 1964654/810665, 786674809783/286794631705, ...

Cf. A229557 (denominators).
Rows 1-4 are A000012(n), A000027(n), A001008(n), A096987(n+1).
Columns 1-2 are A000012(n), A000045(n+2).
Column 3 gives A350834.

A229556A := proc(n,k)
    option remember;
    if n = 1 then
        1;
    else
        add( 1/procname(n-1,c),c=1..k) ;
    end if;
end proc:
A229556 := proc(n,k)
    numer(A229556A(n,k)) ;
end proc:
for d from 2 to 12 do
    for k from d-1 to 1 by -1 do
        printf("%d,",A229556(d-k,k)) ;
    end do:
end do:

cp's OEIS Frontend

A229556 Array read by antidiagonals. Rows are the numerators of consecutive harmonic transforms starting with a first row 1, 1, 1, ....

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