OFFSET
1,5
COMMENTS
See Ma and Chow (2012) for precise definition (cf. On combinations of polynomials and Euler numbers).
LINKS
Shi-Mei Ma and Chak-On Chow, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264 [math.CO], 2012.
EXAMPLE
Triangle begins:
1
1,1
1,2,1
4,7,5,2
11,28,28,16,5
41,131,153,118,71,16
162,690,872,892,759,272,61
...
MATHEMATICA
P[1] := x y; P[n_] := P[n] = ((n-1) q + x y) P[n-1] + 2 q (1-q) D[P[n-1], q] + x (1-q) D[P[n-1], x] + (1-y) D[P[n-1], y] // Simplify;
V[1] = x y; V[n_] := V[n] = ((n-1) q + x y) V[n-1] + 2 q (1-q) D[V[n-1], q] + 2 x (1-q) D[V[n-1], x] + (1 - 2 y + q y) D[V[n-1], y] // Simplify;
Dn[n_] := P[n] /. {x -> 1, y -> 0};
Dbar[n_] := V[n] /. {x -> 1, y -> 0};
Inq[1] = 1; Inq[n_] := (Dn[n] /. q -> q^2) + q (Dbar[n] /. q -> q^2);
Table[CoefficientList[Inq[n], q], {n, 1, 10}] // Flatten (* Jean-François Alcover, Sep 25 2018 *)
PROG
(PARI) tabl(m) = { J = 1; for (d=0, poldegree(J, q), print1(polcoeff(J, d, q), ", "); ); print(""); Pa = x; Pb = x; Pa1 = subst(Pa, x, 1); Pb1 = subst(Pb, x, 1); J = subst(Pa1, q, q^2) + q*subst(Pb1, q, q^2); for (d=0, poldegree(J, q), print1(polcoeff(J, d, q), ", "); ); print(""); Qa = (1+q)*x; Qb = 2*x; for (n=3, m, Qa1 = subst(Qa, x, 1); Qb1 = subst(Qb, x, 1); J = subst(Qa1, q, q^2) + q*subst(Qb1, q, q^2); for (d=0, poldegree(J, q), print1(polcoeff(J, d, q), ", "); ); print(""); newPa = n*q*Qa + 2*q*(1-q)*deriv(Qa, q) + x*(1-q)*deriv(Qa, x) + n*x*Pa; newPb = n*q*Qb + 2*q*(1-q)*deriv(Qb, q) + 2*x*(1-q)*deriv(Qb, x) + n*x*Pb; Pa = Qa; Qa = newPa; Pb = Qb; Qb = newPb; ); } \\ Michel Marcus, Feb 11 2013
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Sep 27 2012
EXTENSIONS
More terms from Michel Marcus, Feb 11 2013
STATUS
approved