A217204 Triangle read by rows, related to Bell numbers A000110: A216962 interlaced with A216964.
1, 2, 1, 5, 6, 1, 15, 22, 9, 2, 52, 94, 63, 26, 5, 203, 460, 416, 244, 101, 16, 877, 2532, 2741, 2124, 1361, 384, 61, 4140, 15420, 18425, 18536, 15602, 6092, 2153, 272, 21147, 102620, 127603, 166440, 165786, 83436, 46959, 10384, 1385, 115975, 739512, 914508, 1550864, 1700220, 1082712, 823256, 247776, 74841, 7936
Offset: 1
Examples
Triangle begins:
1;
2, 1;
5, 6, 1;
15, 22, 9, 2;
52, 94, 63, 26, 5;
203, 460, 416, 244, 101, 16;
...
Links
- S.-M. Ma, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264 [math.CO], 2012.
Programs
-
Mathematica
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; 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; M[n_] := P[n] /. {x -> 1, y -> 1}; Mbar[n_] := V[n] /. {x -> 1, y -> 1}; R[1]=1; R[2] = 2+q; R[n_] := (M[n] /. q -> q^2) + q (Mbar[n] /. q -> q^2); Table[CoefficientList[R[n], q], {n, 1, 10}] // Flatten (* Jean-François Alcover, Sep 25 2018 *) -
PARI
tabl(m) = {Pa = x; Pb = x*y; for (n=1, m, Pa1 = subst(Pa, x, 1); Pb1 = subst(Pb, x, 1); Pb1 = subst(Pb1, y, 1); if (n==1, R = 1, if (n==2, R = 2+q, R = subst(Pa1, q, q^2) + q*subst(Pb1, q, q^2););); for (d=0, poldegree(R, q), print1(polcoeff(R, d, q), ", "); ); print(""); Pa = (n*q+x)*Pa + 2*q*(1-q)*deriv(Pa, q)+ x*(1-q)*deriv(Pa,x); Pb = (n*q+x*y)*Pb + 2*q*(1-q)*deriv(Pb, q)+ 2*x*(1-q)*deriv(Pb,x)+ (1-2*y+q*y)*deriv(Pb,y););} \\ Michel Marcus, Feb 11 2013
Extensions
Example and tabf keyword corrected, and extended by Michel Marcus, Feb 11 2013
Comments