[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;
ColumnForm[ta=PadRight[#,n+1]&/@
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
-------------------------------------
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