A000958 Number of ordered rooted trees with n edges having root of odd degree.

1, 1, 3, 8, 24, 75, 243, 808, 2742, 9458, 33062, 116868, 417022, 1500159, 5434563, 19808976, 72596742, 267343374, 988779258, 3671302176, 13679542632, 51134644014, 191703766638, 720629997168, 2715610275804, 10256844598900, 38822029694628, 147229736485868

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the number of Dyck n-paths containing no peak at height 2 before the first return to ground level. Example: a(3)=3 counts UUUDDD, UDUUDD, UDUDUD. - David Callan, Jun 07 2006

Also number of order trees with n edges and having no even-length branches starting at the root. - Emeric Deutsch, Mar 02 2007

Convolution of the Catalan sequence 1,1,2,5,14,42,... (A000108) and the Fine sequence 1,0,1,2,6,18,... (A000957). a(n) = A127541(n,0). - Emeric Deutsch, Mar 02 2007

The Catalan transform of A008619. - R. J. Mathar, Nov 06 2008

Hankel transform is F(2n+1). - Paul Barry, Dec 01 2008

Starting with offset 2 = iterates of M * [1,1,0,0,0,...] where M = a tridiagonal matrix with [0,2,2,2,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. - Gary W. Adamson, Jan 09 2009

Equals INVERT transform of A032357. - Gary W. Adamson, Apr 10 2009

a(n) is the number of Dyck paths of semilength n+1 that have equal length inclines incident with the first return to ground level. For example, for UUDDUUDDUD these inclines are DD and UU (steps 3 through 6), and a(3)=3 counts UDUDUUDD, UDUDUDUD, UUDDUUDD. - David Callan, Aug 23 2011

a(n) is the number of imprimitive Dyck paths of semilength n+1 for which the heights of the first and the last peaks coincide, this gives the connection to A193215. - Volodymyr Mazorchuk, Aug 27 2011

a(n) is the number of parking functions of size n-1 avoiding the patterns 123 and 132. - Lara Pudwell, Apr 10 2023

a(n) is the number of Dyck paths of semilength n that contain no UDUs at ground level. For example, a(3) = 3 counts UUUDDD, UUDUDD, UUDDUD. - David Callan, Feb 02 2024

References

• Ki Hang Kim, Douglas G. Rogers, and Fred W. Roush, Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577-594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013) - N. J. A. Sloane, Jun 05 2012
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from T. D. Noe)
• Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
• Paul Barry, Moment sequences, transformations, and Spidernet graphs, arXiv:2307.00098 [math.CO], 2023.
• Paul Barry, The Triple Riordan Group, arXiv:2412.05461 [math.CO], 2024. See pp. 6, 10.
• Dennis E. Davenport, Louis W. Shapiro, and Leon C. Woodson, A bijection between the triangulations of convex polygons and ordered trees, Integers (2020) Vol. 20, Article #A8.
• Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
• Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.
• T. Fine, Extrapolation when very little is known about the source, Information and Control 16 (1970), 331-359.
• D. G. Rogers, Similarity relations on finite ordered sets, J. Combin. Theory, A 23 (1977), 88-98. Erratum, loc. cit., 25 (1978), 95-96.
• Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186. See Table 2.
• Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.
• Index entries for sequences related to rooted trees

Crossrefs

First column of A065602, A098747 and A362563. Row sums of A362563.

Cf. A127541, A127539, A000108, A000957, A032357.

Partial differences give A118973 (for n>=1).

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1-x-(1+x)*Sqrt(1-4*x))/(2*x*(x+2)) )); // G. C. Greubel, Feb 27 2019

Maple

g:=(1-x-(1+x)*sqrt(1-4*x))/2/x/(x+2): gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=1..26); # Emeric Deutsch, Mar 02 2007
A958 := n -> add(binomial(2*n-2*k-2, n-1)*(2*k+1)/n, k=0..floor((n-1)/2)): seq(A958(n), n=1..28); # Johannes W. Meijer, Jul 26 2013
A000958List := proc(m) local A, P, n; A := [1,1]; P := [1,1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-2]]);
A := [op(A), P[-1]] od; A end: A000958List(28); # Peter Luschny, Mar 26 2022
# next Maple program:
b:= proc(n) option remember; `if`(n<3, n*(2-n),
      ((7*n-12)*b(n-1)+(4*n-6)*b(n-2))/(2*n))
    end:
a:= n-> b(n)+b(n+1):
seq(a(n), n=1..32);  # Alois P. Heinz, Apr 26 2023

Mathematica

nn = 30; Rest[CoefficientList[Series[(1-x-(1+x)*Sqrt[1-4*x])/(2*x*(x+2)), {x, 0, nn}], x]] (* T. D. Noe, May 09 2012 *)

PARI

my(x='x+O('x^30)); Vec((1-x-(1+x)*sqrt(1-4*x))/(2*x*(x+2))) \\ G. C. Greubel, Feb 27 2019

Python

from itertools import accumulate
def A000958_list(size):
    if size < 1: return []
    L, accu = [], [1]
    for n in range(size-1):
        accu = list(accumulate(accu+[-accu[-1]]))
        L.append(accu[n])
    return L
print(A000958_list(29)) # Peter Luschny, Apr 25 2016

Python

from itertools import count, islice
def A000958_gen(): # generator of terms
    yield 1
    a, c = 0, 1
    for n in count(1):
        yield (c:=c*((n<<2)+2)//(n+2))+a>>1
        a = c-a>>1
A000958_list = list(islice(A000958_gen(),20)) # Chai Wah Wu, Apr 26 2023

Sage

a=((1-x-(1+x)*sqrt(1-4*x))/(2*x*(x+2))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Feb 27 2019

Formula

a(n) = A000957(n) + A000957(n+1).

G.f.: (1-x-(1+x)*sqrt(1-4*x))/(2*x*(x+2)). - Paul Barry, Jan 26 2007

G.f.: z*C/(1-z^2*C^2), where C=(1-sqrt(1-4*z))/(2*z) is the Catalan function. - Emeric Deutsch, Mar 02 2007

a(n+1) = Sum_{k=0..floor(n/2)} A039599(n-k,k). - Philippe Deléham, Mar 13 2007

a(n) = (-1/2)^n*(-2 - 5*Sum_{k=1..n-1} (-8)^k*Gamma(1/2+k)*(4/5+k)/(sqrt(Pi)*Gamma(k+3))). - Mark van Hoeij, Nov 11 2009

a(n) + a(n+1) = A135339(n+1). - Philippe Deléham, Dec 02 2009

From Gary W. Adamson, Jul 14 2011: (Start)

a(n) = sum of top row terms in M^(n-1), where M = the following infinite square production matrix:

0, 1, 0, 0, 0, 0, ...

1, 1, 1, 0, 0, 0, ...

1, 1, 1, 1, 0, 0, ...

1, 1, 1, 1, 1, 0, ...

1, 1, 1, 1, 1, 1, ...

... (End)

D-finite with recurrence 2*(n+1)*a(n) + (-5*n+3)*a(n-1) + (-11*n+21)*a(n-2) + 2 *(-2*n+5)*a(n-3) = 0. - R. J. Mathar, Dec 03 2012

a(n) ~ 5*4^n/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 09 2013

a(n) = Catalan(n-1)*h(n-1) for n>=2 where h(n) = hypergeom([1,3/2,-n/2,(1-n)/2],[1/2,-n,-n+1/2], 1). - Peter Luschny, Apr 25 2016