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

1,3

Comments

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

Links

Table of n, a(n) for n=1..39.

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

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 *)

Prog

(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

Crossrefs

First column is A217203.

Sequence in context: A281897 A282106 A282260 * A100489 A176379 A282454

Adjacent sequences: A217199 A217200 A217201 * A217203 A217204 A217205

Keyword

nonn,tabf

Author

N. J. A. Sloane, Sep 27 2012

Extensions

More terms from Michel Marcus, Feb 09 2013

Status

approved