OFFSET
0,3
COMMENTS
FORMULA
E.g.f.: G(t,z) = exp[(1/4)(1-t)(2z-1+exp(2z))]/(1-z)^t.
The 4-variate g.f. H(u,v,w,z) (exponential with respect z), where u marks number of cycles with 1 alternating run, v marks number of cycles with 2 alternating runs, w marks the number of all cycles, and z marks the size of the permutation, is given by H(u,v,w,z) = exp[(1/4)w((v-1)(exp(2z)+2z)+4(u-v)exp(z)+1-4u+3v)]/(1-z)^w.
We have G(t,z) = H(1/t,1/t,t,z).
EXAMPLE
T(4,1)=2 because we have (1324) and (1423).
Triangle starts:
1;
1;
2;
6;
22,2;
94,26;
460,260;
MAPLE
G := exp((1/4*(1-t))*(2*z-1+exp(2*z)))/(1-z)^t: Gser := simplify(series(G, z = 0, 17)): for n from 0 to 14 do P[n] := sort(factorial(n)*coeff(Gser, z, n)) end do: for n from 0 to 14 do seq(coeff(P[n], t, k), k = 0 .. floor((1/4)*n)) end do; # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Mar 08 2011
STATUS
approved