A128730 -id:A128730 - cp's OEIS frontend

A128728 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDL's (n >= 0; 0 <= k <= floor((n+1)/2)).

1, 1, 2, 1, 6, 4, 20, 16, 71, 64, 2, 262, 261, 20, 994, 1084, 141, 3852, 4572, 854, 7, 15183, 19520, 4772, 112, 60686, 84139, 25416, 1128, 245412, 365404, 131270, 9120, 30, 1002344, 1596420, 664004, 64790, 660, 4129012, 7008544, 3309336, 422928

Offset: 0

Emeric Deutsch, Mar 31 2007

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.
Row n has 1 + floor((n+1)/3) terms.
Row sums yield A002212.
T(n,0) = A128729(n).
Sum_{k>=0} k*T(n,k) = A128730(n).
Apparently, T(3k-1,k) = binomial(3k-1,k)/(3k-1) = A006013(k-1).

			T(3,1)=4 because we have UDUUDL, UUUDLD, UUDUDL and UUUDLL.
Triangle starts:
    1;
    1;
    2,   1;
    6,   4;
   20,  16;
   71,  64,   2;
  262, 261,  20;

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.
Yuxuan Zhang and Yan Zhuang, A subfamily of skew Dyck paths related to k-ary trees, arXiv:2306.15778 [math.CO], 2023.

Cf. A002212, A006013, A128729, A128730.

eq:=z^2*G^3-z*(2-z)*G^2+(1-z^2)*G-1+z+z^2-t*z^2=0: G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor((n+1)/3)) od; # yields sequence in triangular form

G.f.: G = G(t,z) satisfies z^2*G^3 - z(2-z)G^2 + (1-z^2)G - 1 + z + z^2 - tz^2 = 0.

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

cp's OEIS Frontend

A128728 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k UDL's (n >= 0; 0 <= k <= floor((n+1)/2)).

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