# [seqfan] Re: funny recursion producing a known table

Paul D Hanna pauldhanna at juno.com
Sat Apr 27 02:15:21 CEST 2013

```Wouter (and SeqFans),
For the Mma-less (such as myself), your triangle may be expressed also by:

(PARI) {T(n,k)=if(n>=k,if(n==k,1,if((n-k+1)%2==0,0,if(k==0,sum(m=0, n,T(n\2,m)),T(n-1,k-1)+T(n-2*k,k)))))}
for(n=0,20,for(k=0,n,print1(T(n,k),", "));print(""))

Another formula to add to your triangle:

A066897(n) = Sum_{k=0..n} k*T(n,k), where A066897(n) = total number of odd parts in all partitions of n. .

I'm sure there are other such latent formulas.
Paul

---------- Original Message ----------
From: "Wouter Meeussen" <wouter.meeussen at telenet.be>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] funny recursion producing a known table
Date: Fri, 26 Apr 2013 23:52:24 +0200

hi All,

try this one out:

Clear[T];
T[0, 0]:= 1;
T[n_, k_]:= 0 /; ( n<= 0|| OddQ[n +k] ); (* aerated triangular table *)
T[n_, k_]:=T[n, k]=  If[ k==0,   Sum[ T[ n/2, m],{m, 0, n}],  T[n -1, k -1]+
T[ n- 2*k ,k] ];

(* and then show it using *)  Table[T[n, k], {n, 0, 12}, {k, 0, n}]

The fun is in the surprising row sums and the familiar columns down to the
diagonal :
why does such a silly  "T[n -1, k -1]+ T[ n- 2*k ,k] " produce this cute
result (from Wolfdieter Lang)?

And what on earth does it have to do with its GF? (from Emeric Deutsch)
GF :  1/ Product[ (1-t*x^(2*j-1)) * (1-x^(2*j) )

hope the Mathematica code is readable to all,
enjoy,

Wouter.

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