[seqfan] funny recursion producing a known table
Wouter Meeussen
wouter.meeussen at telenet.be
Fri Apr 26 23:52:24 CEST 2013
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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