# FW: catalans : not yet an other interesting property?

Wouter Meeussen w.meeussen.vdmcc at vandemoortele.be
Wed Mar 14 19:46:10 CET 2001

to any sequence A(n) do: Sum( 1/(A(1)+t) + 1/(A(2)+t) + ...1/(A(i)+t), i= 0 .. m )
and verify that the denominator is Product[A(i) + t,{i,0,m}].

Now look at numerator only, and introduce a sign oscillation by replacing t with (-t):
CoefficientList[Sum[1/(f[k]- t),{k,0,m}] Product[f[k]-t,{k,0,m}]//Simplify ,t]

What you get is a triangular table of coefficients:

{1}
{A[0] + A[1], -2}
{A[1]*A[2] + A[0]*(A[1] + A[2]), -2*(A[0] + A[1] + A[2]), 3}
{4*A[0]*A[1]*A[2]*A[3], -4*(A[1]*A[2]*A[3] + A[0]*(A[2]*A[3]
+ A[1]*(A[2] + A[3]))), 4*(A[2]*A[3] + A[1]*(A[2] + A[3])
+ A[0]*(A[1] + A[2] + A[3])), -4*(A[0] + A[1] + A[2] + A[3]), 4}

the interesting bit starts here:

choose the catalans as sequence A(n) and you get:

{1}
{2, -2}
{5, -8, 3}
{27, -50, 27, -4}
{388, -754, 453, -92, 5}
{16436, -32444, 20157, -4468, 325, -6}
{2175432, -4315480, 2709390, -616652, 48485, -1182, 7}
{934036488, -1855691784, 1168801530, -268156228, 21570880, -565260, 4382, -8}
{1336005150480, -2655507324096, 1674169725576, -385021808080, 31181553685,
-834206856, 6925730, -16448, 9}
{6496133192508960, -12913748620055712, 8143796466736656, -1874208257185728,
152085991276570, -4093331598294, 34646140592, -87885296, 62262, -10}

and yes, each row sums to zero except for row 1.
This does not (at first sight) simplify to the well known
Sum[A(k) A(n-k)  , {k,0,n} ] == A(n+1)

is it however new?

Wouter Meeussen
tel +32 (0)51 33 21 24
fax +32 (0)51 33 21 75
wouter.meeussen at vandemoortele.com