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A217202 Triangle read by rows, arising in enumeration of permutations by cyclic valleys, cycles and fixed points. 1
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified February 18 09:15 EST 2020. Contains 332011 sequences. (Running on oeis4.)