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A216963 Triangle read by rows, arising in enumeration of permutations by cyclic peaks, cycles and fixed points.

%I A216963 #31 Nov 08 2017 10:29:09
%S A216963 1,0,1,1,1,4,5,11,28,5,41,153,71,162,872,759,61,715,5191,7262,1665,
%T A216963 3425,32398,66510,29778,1385,17722,211937,601080,443231,60991,98253,
%U A216963 1451599,5446847,5994473,1642877,50521,580317,10393114,49940615,76889330,35162440,3249025
%N A216963 Triangle read by rows, arising in enumeration of permutations by cyclic peaks, cycles and fixed points.
%C A216963 See Ma and Chow (2012) for precise definition (cf. Proposition 3).
%H A216963 Alois P. Heinz, <a href="/A216963/b216963.txt">Rows n = 0..200, flattened</a>
%H A216963 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 A216963 Triangle begins:
%e A216963 :   1;
%e A216963 :   0;
%e A216963 :   1;
%e A216963 :   1,    1;
%e A216963 :   4,    5;
%e A216963 :  11,   28,    5;
%e A216963 :  41,  153,   71;
%e A216963 : 162,  872,  759,   61;
%e A216963 : 715, 5191, 7262, 1665;
%e A216963 ...
%p A216963 p:= proc(n) option remember; expand(`if`(n<4,
%p A216963       [1, 0, x, x*(1+q)][n+1], (n-1)*q*p(n-1)+
%p A216963       2*q*(1-q)*diff(p(n-1), q)+x*(1-q)*
%p A216963       diff(p(n-1), x)+(n-1)*x*p(n-2)))
%p A216963     end:
%p A216963 T:= n-> (t-> seq(coeff(t, q, i), i=0..
%p A216963          max(0, degree(t))))(subs(x=1, p(n))):
%p A216963 seq(T(n), n=0..15);  # _Alois P. Heinz_, Apr 13 2017
%t A216963 p[0] = 1; p[1] = 0; p[2] = x; p[3] = (1 + q) x;
%t A216963 p[n_] := p[n] = Expand[(n - 1) q p[n - 1] + 2 q (1 - q) D[p[n - 1], q] + x (1 - q) D[p[n - 1], x] + (n - 1) x p[n - 2]];
%t A216963 T[n_] := CoefficientList[p[n] /. x -> 1 , q]; T[1] = {0};
%t A216963 Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Nov 08 2017 *)
%o A216963 (PARI) tabf(m) = {P = x; M = subst(P, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); Q = (1+q)*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) + 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
%Y A216963 Column k=0 gives A000296.
%Y A216963 Row sums give A000166.
%Y A216963 T(2n+1,n) gives A000364(n) for n>0.
%K A216963 nonn,tabf
%O A216963 0,6
%A A216963 _N. J. A. Sloane_, Sep 27 2012
%E A216963 More terms from _Michel Marcus_, Feb 09 2013
%E A216963 One row for T(0,0)=1 prepended by _Alois P. Heinz_, Apr 13 2017