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

0,3

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

Table of n, a(n) for n=0..13.

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

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 *)

Crossrefs

Sequence in context: A118584 A126185 A083092 * A220548 A153825 A201490

Adjacent sequences: A174416 A174417 A174418 * A174420 A174421 A174422

Keyword

nonn,frac,eigen

Author

Paul Curtz, Mar 19 2010

Status

approved