A217205 - OEIS

%I #18 Sep 25 2018 11:55:40

%S 1,1,1,1,2,1,4,7,5,2,11,28,28,16,5,41,131,153,118,71,16,162,690,872,

%T 892,759,272,61,715,4033,5191,7060,7262,3468,1665,272,3425,25864,

%U 32398,58608,66510,41088,29778,7936,1385,17722,180265,211937,510812,601080,479772,443231,156176,60991,7936

%N Triangle read by rows, related to Bell numbers A000110.

%C See Ma and Chow (2012) for precise definition (cf. On combinations of polynomials and Euler numbers).

%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 [math.CO], 2012.

%e Triangle begins:

%e 1

%e 1,1

%e 1,2,1

%e 4,7,5,2

%e 11,28,28,16,5

%e 41,131,153,118,71,16

%e 162,690,872,892,759,272,61

%e ...

%t P[1] := x y; P[n_] := P[n] = ((n-1) q + x y) P[n-1] + 2 q (1-q) D[P[n-1], q] + x (1-q) D[P[n-1], x] + (1-y) D[P[n-1], y] // Simplify;

%t V[1] = x y; V[n_] := V[n] = ((n-1) q + x y) V[n-1] + 2 q (1-q) D[V[n-1], q] + 2 x (1-q) D[V[n-1], x] + (1 - 2 y + q y) D[V[n-1], y] // Simplify;

%t Dn[n_] := P[n] /. {x -> 1, y -> 0};

%t Dbar[n_] := V[n] /. {x -> 1, y -> 0};

%t Inq[1] = 1; Inq[n_] := (Dn[n] /. q -> q^2) + q (Dbar[n] /. q -> q^2);

%t Table[CoefficientList[Inq[n], q], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Sep 25 2018 *)

%o (PARI) tabl(m) = { J = 1; for (d=0, poldegree(J, q), print1(polcoeff(J, d, q), ", "); ); print(""); Pa = x; Pb = x; Pa1 = subst(Pa, x, 1);  Pb1 = subst(Pb, x, 1); J = subst(Pa1, q, q^2) + q*subst(Pb1, q, q^2); for (d=0, poldegree(J, q), print1(polcoeff(J, d, q), ", "); ); print(""); Qa = (1+q)*x; Qb = 2*x; for (n=3, m, Qa1 = subst(Qa, x, 1); Qb1 = subst(Qb, x, 1); J = subst(Qa1, q, q^2) + q*subst(Qb1, q, q^2); for (d=0, poldegree(J, q), print1(polcoeff(J, d, q), ", "); ); print(""); newPa = n*q*Qa + 2*q*(1-q)*deriv(Qa,q) + x*(1-q)*deriv(Qa,x) + n*x*Pa; newPb = n*q*Qb + 2*q*(1-q)*deriv(Qb,q) + 2*x*(1-q)*deriv(Qb,x) + n*x*Pb; Pa = Qa; Qa = newPa; Pb = Qb; Qb = newPb;);} \\ _Michel Marcus_, Feb 11 2013

%Y First column appears to be A032265.

%K nonn,tabl

%O 1,5

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

%E More terms from _Michel Marcus_, Feb 11 2013