A128728 -id:A128728 - cp's OEIS frontend

A128730 Number of UDL's in all skew Dyck paths of semilength n.

0, 0, 1, 4, 16, 68, 301, 1366, 6301, 29400, 138355, 655424, 3121438, 14930540, 71675839, 345148892, 1666432816, 8064278288, 39103576699, 189949958332, 924163714216, 4502711570988, 21966152501239, 107284324830302

Offset: 0

Emeric Deutsch, Mar 31 2007

nonn

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of steps in it.

			a(3) = 4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDLL (the other six skew Dyck paths of semilength 3 are the five Dyck paths and UUUDDL).

G. C. Greubel, Table of n, a(n) for n = 0..1000
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Cf. A128728, A014300.

G:=2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..26);

CoefficientList[Series[2*x^2/(1-6*x+5*x^2+(1+x)*Sqrt[1-6*x+5*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *)

z='z+O('z^50); concat([0,0], Vec(2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)))) \\ G. C. Greubel, Mar 19 2017

a(n) = Sum_{k>=0} k*A128728(n,k).
G.f.: 2*z^2/(1-6*z+5*z^2+(1+z)*sqrt(1-6*z+5*z^2)).
a(n) ~ 5^(n-1/2)/(6*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
D-finite with recurrence: +2*(n-1)*(3*n-8)*a(n) +(-39*n^2+161*n-148)*a(n-1) +(48*n^2-215*n+220)*a(n-2) -5*(3*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 17 2016
For n >= 2, a(n) = Sum_{k=1..n-1} binomial(n,k)*A014300(k). - Jianing Song, Apr 20 2019

A128729 Number of skew Dyck paths of semilength n with no UDL's.

Original entry on oeis.org

1, 1, 2, 6, 20, 71, 262, 994, 3852, 15183, 60686, 245412, 1002344, 4129012, 17135432, 71575350, 300690836, 1269662127, 5385593406, 22938095326, 98059308676, 420610907183, 1809690341366, 7808145901068, 33776362530776

Offset: 0

Emeric Deutsch, Mar 31 2007

nonn

			a(2)=2 because we have UDUD and UUDD (UUDL does not qualify).

E. Deutsch, E. Munarini, and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
Helmut Prodinger, Skew Dyck paths without up-down-left, arXiv:2203.10516 [math.CO], 2022.

Cf. A128728.

eq:=z^2*G^3-z*(2-z)*G^2+(1-z^2)*G-1+z+z^2=0: G:=RootOf(eq,G): Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=0..27);

a(n) = A128728(n,0).
G.f.: G = G(z) satisfies z^2*G^3 - z(2-z)G^2 + (1 - z^2)G - 1 + z + z^2 = 0.
D-finite with recurrence 4*n*(n+1)*a(n) -32*n*(n-1)*a(n-1) +3*(23*n^2-78*n+59)*a(n-2) -2*(n-3)*(10*n-47)*a(n-3) -44*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 22 2022

A303952 a(n) is the number of monic polynomials P(z) of degree n over the complex numbers such that P(z) divides P(z^2).

Original entry on oeis.org

1, 2, 5, 17, 69, 302, 1367, 6302, 29401, 138356, 655425, 3121439, 14930541, 71675840, 345148893, 1666432817, 8064278289, 39103576700, 189949958333, 924163714217, 4502711570989, 21966152501240, 107284324830303

Offset: 0

Jianing Song, May 03 2018

Note that if z_0 is a root of P(z), so is (z_0)^2, so |z_0| must equal to 0 or 1. As a result, all such polynomials must have the form P(z) = z^d_0 * Product_{j=1..k} (z - exp(2*Pi*i*q_j))^d_j, where Sum_{j=0..k} d_j = n and {q_1, q_2, ..., q_k} is a set of k rational numbers on [0,1) such that if x belongs to it, the fractional part of 2x also belongs to it. That explains the formula a(n) = Sum_{k=1..n} binomial(n,k)*A014300(k) + 1 in the formula section, the "+1" represents the case d_0 = n and k = 0 corresponding to the polynomial P(z) = z^n.

			For n = 0, P(z) = 1.
For n = 1, P(z) = z or z - 1.
For n = 2, P(z) = z^2, z^2 - 1, z^2 - 2z + 1, z^2 + z or z^2 + z + 1.

Cf. A014300, A128728, A128730.

x='x+O('x^50); Vec(2*x/(1-6*x+5*x^2+(1+x)*sqrt(1-6*x+5*x^2))+1/(1-x))

a(n) = Sum_{k=1..n} binomial(n,k)*A014300(k) + 1. The "+1" represents the polynomial P(z) = z^n.
a(n) = A128730(n+1) + 1.
G.f.: 2x/(1-6x+5x^2+(1+x)sqrt(1-6x+5x^2)) + 1/(1-x).
D-finite with recurrence: +2*n*a(n) +(-13*n+4)*a(n-1) +2*(7*n+3)*a(n-2) +8*(n-7)*a(n-3) +2*(-8*n+33)*a(n-4) +5*(n-4)*a(n-5)=0. - R. J. Mathar, Jan 27 2020
D-finite with recurrence 2*n*a(n) +(-11*n+2)*a(n-1) +(3*n+19)*a(n-2) +(11*n-40)*a(n-3) +5*(-n+3)*a(n-4) +4=0. - R. J. Mathar, Aug 01 2022

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A128730 Number of UDL's in all skew Dyck paths of semilength n.

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A128729 Number of skew Dyck paths of semilength n with no UDL's.

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A303952 a(n) is the number of monic polynomials P(z) of degree n over the complex numbers such that P(z) divides P(z^2).

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