OFFSET
0,8
REFERENCES
Ira. M. Gessel, Generating functions and enumeration of sequences, Ph. D. Thesis, MIT, 1977.
FORMULA
E.g.f.: Bexp(-Ax)/[A*sinh(Bx)+B*cosh(Bx)-sinh(Bx)], where A=(1-t)/2 and B=(1/2)sqrt(t^2+2t-3).
EXAMPLE
Triangle starts:
1;
0,1;
1,0,1;
1,4,0,1;
6,6,11,0,1;
19,51,23,26,0,1
Row n has n+1 terms.
T(3,0)=1, T(3,1)=4, T(3,2)=0 and T(3,3)=1 because we have 123, 13(2), (2)13, 23(1), (3)12, (3)(2)(1), the runs of length 1 being shown between parentheses.
MAPLE
A:=(1-t)/2: B:=sqrt(t^2+2*t-3)/2: G:=B/exp(A*z)/(A*sinh(B*z)+B*cosh(B*z)-sinh(B*z)): Gserz:=simplify(series(G, z=0, 12)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(n!*coeff(Gserz, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..n+1), n=0..10);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch and Ira M. Gessel, Sep 03 2004
STATUS
approved