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A216964 Triangle read by rows, arising in enumeration of permutations by cyclic valleys, cycles and fixed points.

%I A216964 #23 Sep 23 2018 08:57:14
%S A216964 1,2,6,22,2,94,26,460,244,16,2532,2124,384,15420,18536,6092,272,
%T A216964 102620,166440,83436,10384,739512,1550864,1082712,247776,7936,5729192,
%U A216964 15040112,13841928,4864480,441088,47429896,151960264,177512632,87003032,14741984,353792
%N A216964 Triangle read by rows, arising in enumeration of permutations by cyclic valleys, cycles and fixed points.
%C A216964 See Ma and Chow (2012) for precise definition (see Corollary 5).
%H A216964 Shi-Mei Ma and Chak-On Chow, <a href="https://arxiv.org/abs/1203.6264">Enumeration of permutations by number of cyclic peaks and cyclic valleys</a>, arXiv preprint arXiv:1203.6264 [math.CO], 2012.
%e A216964 Triangle begins:
%e A216964 1
%e A216964 2
%e A216964 6
%e A216964 22, 2
%e A216964 94, 26
%e A216964 460, 244, 16
%e A216964 2532, 2124, 384
%e A216964 ...
%t A216964 rows = 12;
%t A216964 Reap[For[P = x*y; n = 1; Sow[{1}], n < rows, n++, P = (n*q + x*y)*P + 2*q*(1-q)*D[P, q] + 2*x*(1-q)*D[P, x] + (1-2*y+q*y)*D[P, y] // Simplify; Sow[CoefficientList[P /. {x -> 1, y -> 1}, q]]]][[2, 1]] // Flatten (* _Jean-François Alcover_, Sep 23 2018, from PARI *)
%o A216964 (PARI) tabf(m) = {P = x*y; for (n=1, m, M = subst(P, x, 1); M = subst(M, y, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); P = (n*q+x*y)*P + 2*q*(1-q)*deriv(P, q)+ 2*x*(1-q)*deriv(P,x)+ (1-2*y+q*y)*deriv(P,y););} \\ _Michel Marcus_, Feb 08 2013
%Y A216964 First column is A187251.
%K A216964 nonn,tabf
%O A216964 1,2
%A A216964 _N. J. A. Sloane_, Sep 27 2012
%E A216964 More terms from _Michel Marcus_, Feb 08 2013