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Array read by antidiagonals. Rows are the numerators of consecutive harmonic transforms starting with a first row 1, 1, 1, ....
3

%I #26 Jun 13 2022 03:03:30

%S 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,

%T 7,49,353777,1964654,12657,21,1,1,8,363,19595573,786674809783,

%U 14862065179,343693,34,1,1,9,761,239046803,17003676861538314284,13379715149864207035877,35955580499839

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

%C 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).

%e Table begins

%e 1, 1, 1, 1, ...

%e 1, 2, 3, 4, ...

%e 1, 3, 11, 25, ...

%e 1, 5, 73, 2221, ...

%e 1, 8, 749, 1964654, ...

%e which are the numerators of

%e 1, 1, 1, 1, 1, ...

%e 1, 2, 3, 4, 5, ...

%e 1, 3/2, 11/6, 25/12, 137/60, ...

%e 1, 5/3, 73/33, 2221/825, 353777/113025, ...

%e 1, 8/5, 749/365, 1964654/810665, 786674809783/286794631705, ...

%p A229556A := proc(n,k)

%p option remember;

%p if n = 1 then

%p 1;

%p else

%p add( 1/procname(n-1,c),c=1..k) ;

%p end if;

%p end proc:

%p A229556 := proc(n,k)

%p numer(A229556A(n,k)) ;

%p end proc:

%p for d from 2 to 12 do

%p for k from d-1 to 1 by -1 do

%p printf("%d,",A229556(d-k,k)) ;

%p end do:

%p end do:

%Y Cf. A229557 (denominators).

%Y Rows 1-4 are A000012(n), A000027(n), A001008(n), A096987(n+1).

%Y Columns 1-2 are A000012(n), A000045(n+2).

%Y Column 3 gives A350834.

%K nonn,tabl,frac

%O 1,5

%A _Franz Vrabec_, Sep 26 2013