# 1,6,30,140,630,2772,...

Michael Somos somos at grail.cba.csuohio.edu
Wed Jan 26 02:15:13 CET 2000

```Dear SeqFans,

Here is an example of something that comes up regularly in my
sequence searches. There is a sequence

%S A002457 1,6,30,140,630,2772,12012,51480,218790,923780,3879876,16224936,
%N A002457 (2n+2)!/(2*n!*(n+1)!).

with a simple explicit formula. However, this is not the only place these
terms appear. Consider the sequence

%S A046210 1,6,30,140,630,2772,12012,51480,218790,923780,3879876,16224936,
%N A046210 Denominator of central elements of Leibniz's Harmonic Triangle.

which agrees with the formula for the terms listed. Why was this duplication
not noticed before? More puzzling is this sequence

%S A034276 6,30,140,630,2772,12012,51480,218790,923780,3879876,16224936,67603900,
%T A034276 280816200,1163381400,4508102925,17166148785,62801114865,226737261975
%N A034276 a(n)=T(n,n-3), where T is given by A034261.
%A A034276 Clark Kimberling, ck6 at cedar.evansville.edu

which agrees with the other two up to and including term 1163381400. After
that the terms suddenly are different. However, I think it is obvious that
this is a mistake in calculation. The corrected calculation (in PARI) taken
from the formula in A034261 is

T(h,k)= if(k<-1,-(k==-2)*(h>1), C(h+k,k+1) + sum(i=1,h-1,\
sum(j=1,k+1, C(i+j-1,j)*C(h+k-i-j,k-j+1) )))

And now T(n,n-3)=A002457(n-2) is the relationship between these sequences.
I suggest elimination of duplicate sequences and double checking before the
sequence is entered into EIS. Please remember to check. Shalom, Michael

--
Michael Somos <somos at grail.cba.csuohio.edu>     Cleveland State University
http://grail.cba.csuohio.edu/~somos/            Cleveland, Ohio, USA 44115

```