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A216962
Triangle read by rows, arising in enumeration of permutations by cyclic peaks.
1
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
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
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Sep 27 2012
EXTENSIONS
More terms from Michel Marcus, Feb 08 2013
STATUS
approved