OFFSET
1,2
COMMENTS
See Ma (2012) for precise definition (cf. On combinations of polynomials and Euler numbers).
LINKS
S.-M. Ma, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264 [math.CO], 2012.
EXAMPLE
Triangle begins:
1;
2, 1;
5, 6, 1;
15, 22, 9, 2;
52, 94, 63, 26, 5;
203, 460, 416, 244, 101, 16;
...
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;
M[n_] := P[n] /. {x -> 1, y -> 1};
Mbar[n_] := V[n] /. {x -> 1, y -> 1};
R[1]=1; R[2] = 2+q; R[n_] := (M[n] /. q -> q^2) + q (Mbar[n] /. q -> q^2);
Table[CoefficientList[R[n], q], {n, 1, 10}] // Flatten (* Jean-François Alcover, Sep 25 2018 *)
PROG
(PARI) tabl(m) = {Pa = x; Pb = x*y; for (n=1, m, Pa1 = subst(Pa, x, 1); Pb1 = subst(Pb, x, 1); Pb1 = subst(Pb1, y, 1); if (n==1, R = 1, if (n==2, R = 2+q, R = subst(Pa1, q, q^2) + q*subst(Pb1, q, q^2); ); ); for (d=0, poldegree(R, q), print1(polcoeff(R, d, q), ", "); ); print(""); Pa = (n*q+x)*Pa + 2*q*(1-q)*deriv(Pa, q)+ x*(1-q)*deriv(Pa, x); Pb = (n*q+x*y)*Pb + 2*q*(1-q)*deriv(Pb, q)+ 2*x*(1-q)*deriv(Pb, x)+ (1-2*y+q*y)*deriv(Pb, y); ); } \\ Michel Marcus, Feb 11 2013
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Sep 27 2012
EXTENSIONS
Example and tabf keyword corrected, and extended by Michel Marcus, Feb 11 2013
STATUS
approved