cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A243752 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=U=(1,1) and 0=D=(1,-1); triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 3, 1, 1, 11, 2, 9, 16, 12, 4, 1, 1, 57, 69, 5, 127, 161, 98, 35, 7, 1, 323, 927, 180, 1515, 1997, 1056, 280, 14, 4191, 5539, 3967, 1991, 781, 244, 64, 17, 1, 1, 10455, 25638, 18357, 4115, 220, 1, 20705, 68850, 77685, 34840, 5685, 246, 1
Offset: 0

Views

Author

Alois P. Heinz, Jun 09 2014

Keywords

Examples

			Triangle T(n,k) begins:
: n\k :    0     1     2     3    4    5  ...
+-----+----------------------------------------------------------
:  0  :    1;                                 [row  0 of A131427]
:  1  :    0,    1;                           [row  1 of A131427]
:  2  :    0,    1,    1;                     [row  2 of A090181]
:  3  :    1,    3,    1;                     [row  3 of A001263]
:  4  :    1,   11,    2;                     [row  4 of A091156]
:  5  :    9,   16,   12,    4,   1;          [row  5 of A091869]
:  6  :    1,   57,   69,    5;               [row  6 of A091156]
:  7  :  127,  161,   98,   35,   7,   1;     [row  7 of A092107]
:  8  :  323,  927,  180;                     [row  8 of A091958]
:  9  : 1515, 1997, 1056,  280,  14;          [row  9 of A135306]
: 10  : 4191, 5539, 3967, 1991, 781, 244, ... [row 10 of A094507]

Links

Crossrefs

Columns k=0-10 give: A243754, A243770, A243771, A243772, A243773, A243774, A243775, A243776, A243777, A243778, A243779, or main diagonals of A243753, A243827, A243828, A243829, A243830, A243831, A243832, A243833, A243834, A243835, A243836.
Row sums give A000108.
Cf. A098978, A114463, A114848, A116424, A135305, A242450, A243366, A243838.

A108863 Number of Dyck paths containing exactly one UUUD.

Original entry on oeis.org

0, 0, 0, 1, 5, 21, 78, 274, 927, 3061, 9933, 31824, 100972, 317942, 995088, 3099105, 9612735, 29715525, 91595391, 281643480, 864189486, 2646805668, 8093543439, 24713953515, 75370741506, 229604257846, 698754428388, 2124616182139
Offset: 0

Views

Author

David Callan, Jul 25 2005

Keywords

Comments

a(n) = number of Dyck n-paths containing exactly one UUUD.
Conjecture: this is the Motzkin transform of the sequence of three zeros followed by A001651. - R. J. Mathar, Dec 11 2008

Examples

			a(4) = 5 because UUUUDDDD, UUUDUDDD, UUUDDUDD, UDUUUDDD, UUUDDDUD
each contain one UUUD.

Links

Crossrefs

Cf. same as A055219 except for offset and is column k=1 of A091958. Dyck paths containing no UUUD are counted by the Motzkin numbers (A001006).
Column k=8 of A243827.

Programs

  • Mathematica
    CoefficientList[Series[(x-1+(1-2*x)*(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2))/(x*(1-3*x)*(1+x*(1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2))),{x,0,20}],x] (* Vaclav Kotesovec, Mar 22 2014 *)

Formula

G.f. (x-1+(1-2*x)M)/(x(1-3*x)(1+x*M)) = Sum_{n>=0}a(n)x^n where M = (1-x-(1-2*x-3*x^2)^(1/2))/(2*x^2) is the gf for Motzkin numbers (A001006); satisfies z^3 = (1 + z)(1 - 3z)( (1 - 3z + z^2)G + z^2(1 - 3z)G^2 ).
Recurrence: (n-3)*(n+2)*a(n) = (n+1)*(5*n-14)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) - 9*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Mar 22 2014
a(n) ~ 3^n/2 * (1-5*sqrt(3)/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Mar 22 2014

A179176 Number of vertices with even distance from the root in "0-1-2" Motzkin trees on n edges.

Original entry on oeis.org

1, 1, 3, 9, 24, 66, 187, 529, 1506, 4312, 12394, 35742, 103377, 299745, 871011, 2535873, 7395522, 21600720, 63176964, 185004852, 542365407, 1591631595, 4675170690, 13744341390, 40438307599, 119063564395, 350799321531
Offset: 0

Views

Author

Lifoma Salaam, Jan 04 2011

Keywords

Comments

"0,1,2" trees are rooted trees where each vertex has outdegree zero, one, or two. They are counted by the Motzkin numbers.

Examples

			We have a(3)=9, as there are 9 vertices with even distance from the root in the 4 "0-1-2" Motzkin trees on 3 edges.

Links

Crossrefs

Cf. A178834, A121320, A091958, A143364, A091958.

Programs

  • Maple
    with(LREtools): with(FormalPowerSeries): # requires Maple 2022
M:= (1-z-sqrt(1-2*z-3*z^2))/(2*z^2): T:=1/sqrt(1-2*z-3*z^2):
ogf:= (M*T^2)/(2*T-1): req:= FindRE(ogf,z,u(n)):
init:= [1, 1, 3, 9, 24, 66]: iseq:= seq(u(i-1)=init[i],i=1..nops(init)):
rmin:= subs(n=n-4, MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
seq(a(n),n=0..27); # Georg Fischer, Nov 04 2022
# Alternative, using function FindSeq from A174403:
ogf := (1-x-sqrt(-3*x^2-2*x+1))/(2*x^2*(3*x^2+2*sqrt(-3*x^2-2*x+1)+2*x-1)):
a := FindSeq(ogf): seq(a(n), n=0..28); # Peter Luschny, Nov 04 2022

Formula

G.f.: (M*T^2)/(2T-1) where M =(1-z-sqrt(1-2*z-3*z^2))/(2*z^2), the g.f. for the Motzkin numbers, and T=1/sqrt(1-2*z-3*z^2), the g.f. for the central trinomial numbers.
D-finite with recurrence: 3*(n+2)*(2*n-1)*a(n) -(4*n+5)*(2*n-1)*a(n-1) +(-20*n^2-8*n+27)*a(n-2) -3*(2*n+3)*(4*n-3)*a(n-3) -9*(2*n+3)*(n-1)*a(n-4)=0. - R. J. Mathar, Jul 24 2012
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