A174419 Numerators T(0,k) of a top row sequence which generates a signed variant (-1)^n*T(n,0) of itself in the column k=0 under repeated application of the Akiyama-Tanigawa transform.

0, 1, 3, 29, 213, 36361, 5004267, 161159569259, 1604875494550299, 700591444676447407855, 272366765005761133289834097, 441056613421971051554626329901900903, 48264034659082736983682770426524745021503, 162486296853709899698219310156295323853814636455303

Offset: 0

Author

Paul Curtz, Mar 19 2010

Keywords

• nonn
• frac
• eigen

Comments

The sequence contains the numerators of the top row in the following table, where successive rows are constructed by iteration of the Akiyama-Tanigawa transform:

0, 1, 3, 29/5, 213/23,...

-1, -4, -42/5, -1592/115, -55070/2737,..

3, 44/5, 1878/115, 343608/13685, 68612650/1967903,..

-29/5, -1732/115, -360378/13685, -22590376/578795, -74842810298/1416609031,...

213/23, 61708/2737, 74954766/1967903, 2737355924568/49581316085,...

The associated denominators in the first row are 1, 1, 1, 5, 23, 2737, 281129, 7083045155,...

The top row is designed to reproduce itself (up to alternating sign) in the leftmost column under the transformation.

There are other examples of sequences quasi-preserved under the Akiyama-Tanigawa transform: if the first row were T(0,k)= A054977(k), the first column would be identical to the first row (no sign flips in this example).

Another (trivial) example is the all-0 sequence, which produces a table containing only zeros.

Links

• D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Programs

Maple

nmax := 10 ;
T := array(0..nmax,0..nmax) ;
T[0,0] := 0 ; T[0,1] := 1 ; T[1,0] := -1 ;
for n from 2 to nmax do
        T[0,n] := x ;
        for r from 1 to n do k := n-r ; T[r,k] := (k+1)*(T[r-1,k]-T[r-1,k+1]) ;
        end do:
        y := solve( T[n,0] = (-1)^n*T[0,n]) ; T[0,n] :=  y;
        for r from 1 to n do k := n-r ; T[r,k] := (k+1)*(T[r-1,k]-T[r-1,k+1]) ;
        end do:
end do:
seq( numer(T[0,i]),i=0..nmax) ; # R. J. Mathar, Dec 02 2010

Mathematica

nmax=10; t[0,0]=0; t[0,1]=1; t[1,0]=-1; For[n=2, n<= nmax, n++, t[0,n]=x; For[r=1, r<=n, r++, k=n-r; t[r,k]=(k+1)*(t[r-1,k]-t[r-1,k+1]);]; y=x/.Solve[t[n,0]==(-1)^n*t[0,n]]//First; t[0,n]=y; For[r=1, r<=n, r++, k=n-r; t[r,k]=(k+1)*(t[r-1,k]-t[r-1,k+1]);]]; Table[ t[0,i],{i,0,nmax}] // Numerator (* Jean-François Alcover, Sep 18 2012, translated from Maple *)