A216962 - OEIS

%I #16 Sep 25 2018 08:00:13

%S 1,2,5,1,15,9,52,63,5,203,416,101,877,2741,1361,61,4140,18425,15602,

%T 2153,21147,127603,165786,46959,1385,115975,914508,1700220,823256,

%U 74841,678570,6794508,17200589,12802659,2389953,50521,4213597,52354116,173871641,185499899,59207070,3855277

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

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

%H Shi-Mei Ma and Chak-On Chow, <a href="http://arxiv.org/abs/1203.6264">Enumeration of permutations by number of cyclic peaks and cyclic valleys</a>, arXiv preprint arXiv:1203.6264, 2012

%e Triangle begins:

%e 1

%e 2

%e 5,1

%e 15,9

%e 52,63,5

%e 203,416,101

%e 877,2741,1361,61

%e ...

%t 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;

%t row[n_] := CoefficientList[P[n] /. {x -> 1, y -> 1}, q];

%t Array[row, 12] // Flatten (* _Jean-François Alcover_, Sep 25 2018 *)

%o (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

%Y First column is A000110.

%K nonn,tabf

%O 1,2

%A _N. J. A. Sloane_, Sep 27 2012

%E More terms from _Michel Marcus_, Feb 08 2013