A174419 - OEIS

%I #22 Mar 26 2023 14:32:20

%S 0,1,3,29,213,36361,5004267,161159569259,1604875494550299,

%T 700591444676447407855,272366765005761133289834097,

%U 441056613421971051554626329901900903,48264034659082736983682770426524745021503,162486296853709899698219310156295323853814636455303

%N 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.

%C 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:

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

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

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

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

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

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

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

%C 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).

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

%H D. Merlini, R. Sprugnoli, M. C. Verri, <a href="https://www.emis.de/journals/INTEGERS/papers/f5/f5.Abstract.html">The Akiyama-Tanigawa Transformation</a>, Integers, 5 (1) (2005) #A05.

%p nmax := 10 ;

%p T := array(0..nmax,0..nmax) ;

%p T[0,0] := 0 ; T[0,1] := 1 ; T[1,0] := -1 ;

%p for n from 2 to nmax do

%p         T[0,n] := x ;

%p         for r from 1 to n do k := n-r ; T[r,k] := (k+1)*(T[r-1,k]-T[r-1,k+1]) ;

%p         end do:

%p         y := solve( T[n,0] = (-1)^n*T[0,n]) ; T[0,n] :=  y;

%p         for r from 1 to n do k := n-r ; T[r,k] := (k+1)*(T[r-1,k]-T[r-1,k+1]) ;

%p         end do:

%p end do:

%p seq( numer(T[0,i]),i=0..nmax) ; # _R. J. Mathar_, Dec 02 2010

%t 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 *)

%K nonn,frac,eigen

%O 0,3

%A _Paul Curtz_, Mar 19 2010