A216962 Triangle read by rows, arising in enumeration of permutations by cyclic peaks.

1, 2, 5, 1, 15, 9, 52, 63, 5, 203, 416, 101, 877, 2741, 1361, 61, 4140, 18425, 15602, 2153, 21147, 127603, 165786, 46959, 1385, 115975, 914508, 1700220, 823256, 74841, 678570, 6794508, 17200589, 12802659, 2389953, 50521, 4213597, 52354116, 173871641, 185499899, 59207070, 3855277

Offset

1,2

Comments

See Ma and Chow (2012) for precise definition (see corollary 2).

Links

Table of n, a(n) for n=1..42.

Shi-Mei Ma and Chak-On Chow, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264, 2012

Example

Triangle begins:
1
2
5,1
15,9
52,63,5
203,416,101
877,2741,1361,61
...

Mathematica

P[1] := x y; P[n_] := P[n] = ((n-1)q + x y) P[n-1] + 2q(1-q) D[P[n-1], q] + x(1-q) D[P[n-1], x] + (1-y) D[P[n-1], y] // Simplify;
row[n_] := CoefficientList[P[n] /. {x -> 1, y -> 1}, q];
Array[row, 12] // Flatten (* Jean-François Alcover, Sep 25 2018 *)

Prog

(PARI) tabf(m) = {P = x; for (n=1, m, M = subst(P, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); P = (n*q+x)*P + 2*q*(1-q)*deriv(P, q)+ x*(1-q)*deriv(P, x); ); } \\ Michel Marcus, Feb 08 2013

Crossrefs

First column is A000110.

Sequence in context: A110220 A349106 A119518 * A186756 A184940 A185140

Adjacent sequences: A216959 A216960 A216961 * A216963 A216964 A216965

Keyword

nonn,tabf

Author

N. J. A. Sloane, Sep 27 2012

Extensions

More terms from Michel Marcus, Feb 08 2013

Status

approved