A217204 Triangle read by rows, related to Bell numbers A000110.

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

1,2

Comments

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

Links

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

S.-M. Ma, Enumeration of permutations by number of cyclic peaks and cyclic valleys, arXiv preprint arXiv:1203.6264 [math.CO], 2012.

Example

Triangle begins:
    1;
    2,   1;
    5,   6,   1;
   15,  22,   9,   2;
   52,  94,  63,  26,   5;
  203, 460, 416, 244, 101, 16;
  ...

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 *)

Prog

(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

Crossrefs

First column is A000110.

Sequence in context: A095801 A128567 A345394 * A179455 A039810 A328297

Adjacent sequences: A217201 A217202 A217203 * A217205 A217206 A217207

Keyword

nonn,tabl

Author

N. J. A. Sloane, Sep 27 2012

Extensions

Example and tabf keyword corrected, and extended by Michel Marcus, Feb 11 2013

Status

approved