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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. 1
0, 1, 3, 29, 213, 36361, 5004267, 161159569259, 1604875494550299, 700591444676447407855, 272366765005761133289834097, 441056613421971051554626329901900903, 48264034659082736983682770426524745021503, 162486296853709899698219310156295323853814636455303 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,frac,eigen
AUTHOR
Paul Curtz, Mar 19 2010
STATUS
approved

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Last modified July 13 23:31 EDT 2024. Contains 374290 sequences. (Running on oeis4.)