# [seqfan] inverse binomial transform on diagonals of triangular table

wouter meeussen wouter.meeussen at pandora.be
Fri Jul 22 22:43:20 CEST 2011

```a case of  "consistent persistence", maybe familiar to some:

take A009766 (Catalan triangle), the G.F. for the columns are known.

The inverse binomial transform on the columns produces a known triangular
table:
A039599  "Triangle formed from even-numbered columns of triangle of
expansions of powers
of x in terms of Chebyshev polynomials U_n (x)."

Triangle begins:

1;

1, 1;

2, 3, 1;

5, 9, 5, 1;

14, 28, 20, 7, 1;

42, 90, 75, 35, 9, 1;

Now, take the diagonals of that, and again take their inverse Binomial
transform:

a new triangular table results:

1;

1, 2;

2, 7, 4;

5, 23, 24, 8;

14, 76, 109, 68, 16;

42, 255, 449, 394, 176, 32;

...

with properties:

not in OEIS,

columns and diagonals are exponential, no longer polynomial,

first column are the catalans (again),

last row-elements are the powers of 2,

row sums are 1,3,13,60,283,1348,6454,30992,149091,718044,3460818

(nothing on superseeker, hmm..)

but row sums with alternating signs gives a familiar

-1, 1, 1, 2, 5, 14, 42, 132, 429, 1430 ...

or the catalans preceded by a -1.

I propose to enter this table into the OEIS,

on merit of its nice mariage of catalans and powers of 2.

Wouter.

--------- Mma 4.0 ----------------

<<DiscreteMath`Transforms`;

diags[mat_]:=Table[ mat[[i+j-1,j]],{i,First[Dimensions[mat]]} ,
{j,Min[Last[Dimensions[mat]],First[Dimensions[mat]]+1-i]}];

cat[k_,j_]:=(k+j)!(k-j+1)/(k+1)!/j! ;

n=36;
Table[BinomialInvTransform at Table[cat[k,j],{k,j,2n}],{j,0,n}] ];
Dimensions[ta]
diags[ta]
BinomialInvTransform/@ %
Take[ %, n/2+1]
tria=MapIndexed[Take[#1,Tr[#2]]&, %]
ColumnForm[tria]
Tr/@ tria
(# . Table[(-1)^k,{k,Length[#]}])& /@ tria

-------------------------------------

```