A217205 Triangle read by rows, related to Bell numbers A000110: A216963 interlaced with A217202.
1, 1, 1, 1, 2, 1, 4, 7, 5, 2, 11, 28, 28, 16, 5, 41, 131, 153, 118, 71, 16, 162, 690, 872, 892, 759, 272, 61, 715, 4033, 5191, 7060, 7262, 3468, 1665, 272, 3425, 25864, 32398, 58608, 66510, 41088, 29778, 7936, 1385, 17722, 180265, 211937, 510812, 601080, 479772, 443231, 156176, 60991, 7936
Offset: 1
Examples
Triangle begins: 1 1,1 1,2,1 4,7,5,2 11,28,28,16,5 41,131,153,118,71,16 162,690,872,892,759,272,61 ...
Links
- Shi-Mei Ma and Chak-On Chow, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264 [math.CO], 2012.
Programs
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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; Dn[n_] := P[n] /. {x -> 1, y -> 0}; Dbar[n_] := V[n] /. {x -> 1, y -> 0}; Inq[1] = 1; Inq[n_] := (Dn[n] /. q -> q^2) + q (Dbar[n] /. q -> q^2); Table[CoefficientList[Inq[n], q], {n, 1, 10}] // Flatten (* Jean-François Alcover, Sep 25 2018 *) -
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
Extensions
More terms from Michel Marcus, Feb 11 2013
Comments