|
%I #26 Sep 23 2018 22:28:38
%S 0,1,2,7,2,28,16,131,118,16,690,892,272,4033,7060,3468,272,25864,
%T 58608,41088,7936,180265,510812,479772,156176,7936,1354458,4675912,
%U 5635224,2665184,353792,10898823,44918110,67238764,42832648,9972704,353792,93407828,452104928
%N Triangle read by rows, arising in enumeration of permutations by cyclic valleys, cycles and fixed points.
%C See Ma (2012) for precise definition (cf. Proposition 6).
%H S.-M. Ma, <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 Triangle begins:
%e 0;
%e 1;
%e 2;
%e 7, 2;
%e 28, 16;
%e 131, 118, 16;
%e 690, 892, 272;
%e ...
%t V[0][_, _] = 1; V[1][_, _] = 0; V[2][_, x_] := x; V[3][_, x_] := 2x;
%t V[n_][q_, x_] := V[n][q, x] = (n-1) q V[n-1][q, x] + 2q(1-q) D[V[n-1][q, x], q] + 2x (1-q) D[V[n-1][q, x], x] + (n-1) x V[n-2][q, x] // Simplify;
%t Table[If[n==1, {0}, CoefficientList[V[n][q, x] /. x -> 1, q]], {n, 1, 13}] // Flatten (* _Jean-François Alcover_, Sep 23 2018 *)
%o (PARI) tabf(m) = {P = x; M = subst(P, x, 1); for (d=0, poldegree(M, q), print1(polcoeff(M, d, q), ", "); ); print(""); Q = 2*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) + 2*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 First column is A217203.
%K nonn,tabf
%O 1,3
%A _N. J. A. Sloane_, Sep 27 2012
%E More terms from _Michel Marcus_, Feb 09 2013
|
