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A217204 Triangle read by rows, related to Bell numbers A000110. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A120986 A095801 A128567 * 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

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Last modified February 25 19:39 EST 2021. Contains 341618 sequences. (Running on oeis4.)