OFFSET
0,3
COMMENTS
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
The row generating polynomial P[n](u) is given by P[n](u)=Q[n](u,1,1), where Q[n](u,v,w) is obtained recursively from Q[n](u,v,w) =u(dQ[n-1]/du)_{w=v} + u(dQ[n-1]/dv)_{w=v} + w(dQ[n-1]/dw) + w(Q[n-1])_{w=v}, Q[0]=1. Here Q[n](u,v,w) is the trivariate generating polynomial of the partitions of {1,2,...,n}, where u marks blocks that do not consist of consecutive integers, v marks blocks consisting of consecutive integers and not ending with n, and w marks blocks consisting of consecutive integers and ending with n.
EXAMPLE
T(4,1)=6 because we have 134-2, 124-3, 14-23, 1-24-3, 14-2-3, and 13-2-4.
Triangle starts:
1;
1;
2,0;
4,1;
8,6,1;
16,25,11;
32,89,77,5;
MAPLE
Q[0] := 1: for n to 12 do Q[n] := expand(u*subs(w = v, diff(Q[n-1], u))+u*subs(w = v, diff(Q[n-1], v))+w*(diff(Q[n-1], w))+w*subs(w = v, Q[n-1])) end do: for n from 0 to 12 do P[n] := sort(expand(subs({v = 1, w = 1}, Q[n]))) end do: for n from 0 to 12 do seq(coeff(P[n], u, j), j = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, May 07 2010
STATUS
approved