A217205 Triangle read by rows, related to Bell numbers A000110.

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

1,5

Comments

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

Links

Table of n, a(n) for n=1..55.

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.

Example

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
...

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 *)

Prog

(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

Crossrefs

First column appears to be A032265.

Sequence in context: A072015 A123242 A322941 * A358566 A139769 A357470

Adjacent sequences: A217202 A217203 A217204 * A217206 A217207 A217208

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 27 2012

Extensions

More terms from Michel Marcus, Feb 11 2013

Status

approved