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

1,5

Comments

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).

Links

Table of n, a(n) for n=1..52.

Example

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, ...

Maple

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:

Crossrefs

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.

Sequence in context: A209631 A309876 A059922 * A159623 A143199 A137896

Adjacent sequences: A229553 A229554 A229555 * A229557 A229558 A229559

Keyword

nonn,tabl,frac

Author

Franz Vrabec, Sep 26 2013

Status

approved