Induction proof of harmonic triangle formula

[timothychoi]

Hello. In the Wiki entry for "Leibniz harmonic triangle" there one
can find a formula

L(r, c) = 1/(c * binom(r - 1, c - 1))

where L(r, c) is the entry in the harmonic triangle and the
binom() is the usual binomial notation (or the entry from
Pascal triangle). I tried to prove the assertion using indunction
on n, then I was not able to finish. Can someone prove
the assertion using induction on n? Only the last step, that
is assuming that

L(k, c) = 1/(c * binom(k - 1, c - 1))

and showing that

L(k + 1, c) = 1/(c * binom((k + 1) - 1, c - 1)),

will be sufficient. Thank you.

 

[awkward]

Hello. In the Wiki entry for "Leibniz harmonic triangle" there one
can find a formula

L(r, c) = 1/(c * binom(r - 1, c - 1))

where L(r, c) is the entry in the harmonic triangle and the
binom() is the usual binomial notation (or the entry from
Pascal triangle). I tried to prove the assertion using indunction
on n, then I was not able to finish. Can someone prove
the assertion using induction on n? Only the last step, that
is assuming that

L(k, c) = 1/(c * binom(k - 1, c - 1))

and showing that

L(k + 1, c) = 1/(c * binom((k + 1) - 1, c - 1)),

will be sufficient. Thank you.

I think that should be

L(r, c) = 1/(r * binom(r - 1, c - 1))

-evidently an error on Wikipedia. Compare the entry on Mathworld:

http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html