A217202 Triangle read by rows, arising in enumeration of permutations by cyclic valleys, cycles and fixed points.

0, 1, 2, 7, 2, 28, 16, 131, 118, 16, 690, 892, 272, 4033, 7060, 3468, 272, 25864, 58608, 41088, 7936, 180265, 510812, 479772, 156176, 7936, 1354458, 4675912, 5635224, 2665184, 353792, 10898823, 44918110, 67238764, 42832648, 9972704, 353792, 93407828, 452104928

Offset: 1

Author

N. J. A. Sloane, Sep 27 2012

Keywords

• nonn
• tabf

Comments

See Ma (2012) for precise definition (cf. Proposition 6).

Examples

			Triangle begins:
    0;
    1;
    2;
    7,   2;
   28,  16;
  131, 118,  16;
  690, 892, 272;
  ...
		

Links

• S.-M. Ma, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264 [math.CO], 2012.

Crossrefs

First column is A217203.

Programs

Mathematica

V[0][, ] = 1; V[1][, ] = 0; V[2][, x] := x; V[3][, x] := 2x;
V[n_][q_, x_] := V[n][q, x] = (n-1) q V[n-1][q, x] + 2q(1-q) D[V[n-1][q, x], q] + 2x (1-q) D[V[n-1][q, x], x] + (n-1) x V[n-2][q, x] // Simplify;
Table[If[n==1, {0}, CoefficientList[V[n][q, x] /. x -> 1, q]], {n, 1, 13}] // Flatten (* Jean-François Alcover, Sep 23 2018 *)

PARI

tabf(m) = {P = x; M = subst(P, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); Q = 2*x; M = subst(Q, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); for (n=3, m, newP = n*q*Q + 2*q*(1-q)*deriv(Q,q) + 2*x*(1-q)*deriv(Q,x) + n*x*P; M = subst(newP, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); P = Q; Q = newP;);} \\ Michel Marcus, Feb 09 2013

Extensions

More terms from Michel Marcus, Feb 09 2013