A198951 -id:A198951 - cp's OEIS frontend

A036765 Number of ordered rooted trees with n non-root nodes and all outdegrees <= three.

1, 1, 2, 5, 13, 36, 104, 309, 939, 2905, 9118, 28964, 92940, 300808, 980864, 3219205, 10626023, 35252867, 117485454, 393133485, 1320357501, 4449298136, 15038769672, 50973266380, 173214422068, 589998043276, 2014026871496, 6889055189032, 23608722350440

Offset: 0

N. J. A. Sloane

Number of Dyck n-paths that avoid UUUU. For example, a(4)=13 counts all 14 Dyck 4-paths except UUUUDDDD. - David Callan, Dec 09 2004
Number of restricted growth strings for Dyck paths with at most 2 consecutive rises (this is equivalent to the comment above, see example). - Joerg Arndt, Oct 31 2012
Let A(x) be the g.f. for the sequence of numbers of Dyck words with at most k consecutive ones (paths with at most k consecutive up-steps 'U', Restricted Growth Strings with at most k-1 consecutive rises), then B(x) := x*A(x) is the series reversion of x/(1+x+x^2+...+x^k). - Joerg Arndt, Oct 31 2012
a(n) is the number of ordered unlabeled rooted trees on n+1 nodes where each node has no more than 3 children. - Geoffrey Critzer, Jan 05 2013
a(n) = number of noncrossing partitions of [n] in which all blocks are of size <= 3. - David Callan, Aug 27 2014

			a(4) = 13 since the top row of M^4 = (13, 8, 4, 1, 1).
From _Joerg Arndt_, Oct 31 2012: (Start)
a(5)=36 because there are 36 Dyck words of length 5 that avoid "1111":
[ #]      RGS                rises         Dyck word
[ 1]    [ . . . . . ]     [ . . . . . ]    1.1.1.1.1.
[ 2]    [ . . . . 1 ]     [ . . . . 1 ]    1.1.1.11..
[ 3]    [ . . . 1 . ]     [ . . . 1 . ]    1.1.11..1.
[ 4]    [ . . . 1 1 ]     [ . . . 1 . ]    1.1.11.1..
[ 5]    [ . . . 1 2 ]     [ . . . 1 2 ]    1.1.111...
[ 6]    [ . . 1 . . ]     [ . . 1 . . ]    1.11..1.1.
[ 7]    [ . . 1 . 1 ]     [ . . 1 . 1 ]    1.11..11..
[ 8]    [ . . 1 1 . ]     [ . . 1 . . ]    1.11.1..1.
[ 9]    [ . . 1 1 1 ]     [ . . 1 . . ]    1.11.1.1..
[10]    [ . . 1 1 2 ]     [ . . 1 . 1 ]    1.11.11...
[11]    [ . . 1 2 . ]     [ . . 1 2 . ]    1.111...1.
[12]    [ . . 1 2 1 ]     [ . . 1 2 . ]    1.111..1..
[13]    [ . . 1 2 2 ]     [ . . 1 2 . ]    1.111.1...
[--]    [ . . 1 2 3 ]     [ . . 1 2 3 ]    1.1111....
[14]    [ . 1 . . . ]     [ . 1 . . . ]    11..1.1.1.
[15]    [ . 1 . . 1 ]     [ . 1 . . 1 ]    11..1.11..
[16]    [ . 1 . 1 . ]     [ . 1 . 1 . ]    11..11..1.
[17]    [ . 1 . 1 1 ]     [ . 1 . 1 . ]    11..11.1..
[18]    [ . 1 . 1 2 ]     [ . 1 . 1 2 ]    11..111...
[19]    [ . 1 1 . . ]     [ . 1 . . . ]    11.1..1.1.
[20]    [ . 1 1 . 1 ]     [ . 1 . . 1 ]    11.1..11..
[21]    [ . 1 1 1 . ]     [ . 1 . . . ]    11.1.1..1.
[22]    [ . 1 1 1 1 ]     [ . 1 . . . ]    11.1.1.1..
[23]    [ . 1 1 1 2 ]     [ . 1 . . 1 ]    11.1.11...
[24]    [ . 1 1 2 . ]     [ . 1 . 1 . ]    11.11...1.
[25]    [ . 1 1 2 1 ]     [ . 1 . 1 . ]    11.11..1..
[26]    [ . 1 1 2 2 ]     [ . 1 . 1 . ]    11.11.1...
[27]    [ . 1 1 2 3 ]     [ . 1 . 1 2 ]    11.111....
[28]    [ . 1 2 . . ]     [ . 1 2 . . ]    111...1.1.
[29]    [ . 1 2 . 1 ]     [ . 1 2 . 1 ]    111...11..
[30]    [ . 1 2 1 . ]     [ . 1 2 . . ]    111..1..1.
[31]    [ . 1 2 1 1 ]     [ . 1 2 . . ]    111..1.1..
[32]    [ . 1 2 1 2 ]     [ . 1 2 . 1 ]    111..11...
[33]    [ . 1 2 2 . ]     [ . 1 2 . . ]    111.1...1.
[34]    [ . 1 2 2 1 ]     [ . 1 2 . . ]    111.1..1..
[35]    [ . 1 2 2 2 ]     [ . 1 2 . . ]    111.1.1...
[36]    [ . 1 2 2 3 ]     [ . 1 2 . 1 ]    111.11....
[--]    [ . 1 2 3 . ]     [ . 1 2 3 . ]    1111....1.
[--]    [ . 1 2 3 1 ]     [ . 1 2 3 . ]    1111...1..
[--]    [ . 1 2 3 2 ]     [ . 1 2 3 . ]    1111..1...
[--]    [ . 1 2 3 3 ]     [ . 1 2 3 . ]    1111.1....
[--]    [ . 1 2 3 4 ]     [ . 1 2 3 4 ]    11111.....
(Dots are used for zeros for readability.)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from T. D. Noe)
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 25.
N. T. Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.
Naiomi Cameron and J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, Article #16.6.1.
Colin Defant and Kai Zheng, Stack-Sorting with Consecutive-Pattern-Avoiding Stacks, arXiv:2008.12297 [math.CO], 2020.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv preprint arXiv:1210.6908 [math.CO], 2012.
M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.
Petr Gregor, Torsten Mütze, and Namrata, Combinatorial generation via permutation languages. VI. Binary trees, arXiv:2306.08420 [cs.DM], 2023.
Nickolas Hein and Jia Huang, Modular Catalan Numbers, arXiv:1508.01688 [math.CO], 2015.
Nickolas Hein and Jia Huang, Modular Catalan Numbers, European Journal of Combinatorics 61 (2017), 197-218.
JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, Bijections on pattern avoiding inversion sequences and related objects, arXiv:2404.04091 [math.CO], 2024. See p. 22.
L. Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012. - From _N. J. A. Sloane_, Jan 03 2013
A. Sapounakis, I. Tasoulas, and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.
L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (6).
M. Wallner, Lattice Path Combinatorics, Diplomarbeit, Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien, 2013.
Index entries for sequences related to rooted trees

Right hand column of triangle A064580. The sequence of sequences A000007 (0^n), A000012 (constant 1), A001006 (Motzkin), A036765, A036766, ... tends to A000108 (Catalan).
Cf. A005725, A186241, A198951, A200731.
Column k=3 of A288942.

[&+[Binomial(n+1, n-2*k)*Binomial(n+1, k)/(n+1): k in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Oct 16 2018

r := 3; [ seq((1/n)*add( (-1)^j*binomial(n,j)*binomial(2*n-2-j*(r+1), n-1),j=0..floor((n-1)/(r+1))), n=1..30) ];
# second Maple program:
b:= proc(u, o) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1), j=1..min(1, u))+
      add(b(u+j-1, o-j), j=1..min(3, o)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 28 2017

InverseSeries[Series[y/(1+y+y^2+y^3), {y, 0, 24}], x] (* then A(x)=y(x)/x *) (* Len Smiley, Apr 11 2000 *)
b[u_, o_, k_] := b[u, o, k] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1, k], {j, 1, Min[1, u]}] + Sum[b[u + j - 1, o - j, k], {j, 1, Min[k, o]}]];
a[n_] := b[0, n, 3];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 05 2017, after Alois P. Heinz *)
Table[HypergeometricPFQ[{-n-1, (1-n)/2, -n/2}, {1, 3/2}, -1], {n, 0, 28}] (* Vladimir Reshetnikov, Oct 15 2018 *)

{a(n)=sum(j=0,n\2,binomial(n+1, n-2*j)*binomial(n+1,j))/(n+1)}

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*A+(x*A)^2+(x*A)^3);polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(x*A+x*O(x^n))^j)*x^m/m)));polcoeff(A, n, x)}

{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(j=0, n, binomial(m+j, j)^2*(x*A+x*O(x^n))^j)*(1-x*A)^(2*m+1)*x^m/m)));polcoeff(A, n, x)}

{a(n)=local(A=1+x);for(i=1,n,A=1/(1-x+x*O(x^n))*exp(sum(m=1,n,A^m*sum(k=0,m-1,binomial(m-1,k)*binomial(m,k)*x^k)/(1-x)^(2*m)*x^(2*m)/m) +x*O(x^n)));polcoeff(A,n)} /* Paul D. Hanna */

{a(n)=local(A=1+x);for(i=1,n,A=1/(1-x+x*O(x^n))*exp(sum(m=1,n,A^m*sum(k=0,n,binomial(m+k-1,k)*binomial(m+k,k)*x^k)*x^(2*m)/m) +x*O(x^n)));polcoeff(A,n)} /* Paul D. Hanna */

Vec(serreverse(x/(1+x+x^2+x^3)+O(x^66))/x) /* Joerg Arndt, Jun 10 2011 */

a(n) = (1/(n+1))*Sum_{j=0..floor(n/2)} binomial(n+1, n-2*j)*binomial(n+1, j). G.f. A(z) satisfies A=1+z*A+(z*A)^2+(z*A)^3. - Emeric Deutsch, Nov 29 2003
G.f.: F(x)/x where F(x) is the reversion of x/(1+x+x^2+x^3). - Joerg Arndt, Jun 10 2011
From Paul D. Hanna, Feb 13 2011: (Start)
O.g.f.: A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2*x^k*A(x)^k) * x^n/n ).
O.g.f.: A(x) = exp( Sum_{n>=1} (Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^k)*(1-x*A(x))^(2*n+1)* x^n/n ). (End)
From Paul D. Hanna, Feb 24 2011: (Start)
O.g.f.: A(x) = (1/(1-x))*exp( Sum_{n>=1} A(x)^n*(Sum_{k=0..n-1} C(n-1,k)*C(n,k)*x^k)/(1-x)^(2*n) * x^(2*n)/n ).
O.g.f.: A(x) = (1/(1-x))*exp( Sum_{n>=1} A(x)^n*(Sum_{k>=0} C(n+k-1,k)*C(n+k,k)*x^k) * x^(2*n)/n ). (End)
Let M = an infinite quadradiagonal matrix with all 1's in every diagonal: (sub, main, super, and super-super), and the rest zeros. V = vector [1,0,0,0,...]. The sequence = left column terms of M*V iterates. - Gary W. Adamson, Jun 06 2011
An infinite square production matrix M for the sequence is:
  1, 1, 0, 0, 0, 0, ...
  1, 0, 1, 0, 0, 0, ...
  2, 1, 0, 1, 0, 0, ...
  3, 2, 1, 0, 1, 0, ...
  4, 3, 2, 1, 0, 1, ...
  5, 4, 3, 2, 1, 0, ...
  ..., such that a(n) is the top left term of M^n. - Gary W. Adamson, Feb 21 2012
D-finite with recurrence: 2*(n+1)*(2*n+3)*(13*n-1)*a(n) = (143*n^3 + 132*n^2 - 17*n - 18)*a(n-1) + 4*(n-1)*(26*n^2 + 11*n - 6)*a(n-2) + 16*(n-2)*(n-1)*(13*n + 12)*a(n-3). - Vaclav Kotesovec, Sep 09 2013
a(n) ~ c*d^n/n^(3/2), where d = 1/12*((6371+624*sqrt(78))^(2/3)+11*(6371+624*sqrt(78))^(1/3)+217)/(6371+624*sqrt(78))^(1/3) = 3.610718613276... is the root of the equation -16-8*d-11*d^2+4*d^3=0 and c = sqrt(f/Pi) = 0.9102276936417..., where f = 1/9984*(9295 + (13*(45085576939 - 795629568*sqrt(78)))^(1/3) + (13*(45085576939 + 795629568*sqrt(78)))^(1/3)) is the root of the equation -128+1696*f-9295*f^2+3328*f^3=0. - Vaclav Kotesovec, Sep 10 2013
From Peter Bala, Jun 21 2015: (Start)
The coefficient of x^n in A(x)^r equals r/(n + r)*Sum_{k = 0..floor(n/2)} binomial(n + r,k)*binomial(n + r,n - 2*k) by the Lagrange-Bürmann formula.
O.g.f. A(x) = exp(Sum_{n >= 1} A005725(n)*x^n/n), where A005725(n) = Sum_{k = 0..floor(n/2)} binomial(n,k)*binomial(n,n - 2*k). Cf. A186241, A198951, A200731. (End)
a(n) = hypergeom([-n-1, (1-n)/2, -n/2], [1, 3/2], -1). - Vladimir Reshetnikov, Oct 15 2018

Name clarified by Andrew Howroyd, Dec 04 2017

A198953 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^3).

Original entry on oeis.org

1, 2, 9, 56, 400, 3095, 25240, 213633, 1859006, 16527544, 149472480, 1370794835, 12718060947, 119158146283, 1125816405458, 10714275588727, 102615375322564, 988302823695146, 9565859385140272, 93000625498797314, 907782305262566776, 8892941663606408172

Offset: 0

Paul D. Hanna, Oct 31 2011

The radius of convergence of g.f. A(x) is r = 0.095007017562450871521918431664620... with A(r) = 1.6228790124092133906198298670423120590101223122... where y=A(r) satisfies 2*y^5 + 6*y^4 - 18*y^3 + 6*y^2 - 3 = 0.

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 56*x^3 + 400*x^4 + 3095*x^5 + 25240*x^6 +...
Related expansions.
A(x)^2 = 1 + 4*x + 22*x^2 + 148*x^3 + 1105*x^4 + 8798*x^5 + 73196*x^6 +...
A(x)^3 = 1 + 6*x + 39*x^2 + 284*x^3 + 2223*x^4 + 18267*x^5 + 155445*x^6 +...
A(x)^4 = 1 + 8*x + 60*x^2 + 472*x^3 + 3878*x^4 + 32948*x^5 + 287300*x^6 +...
where A(x) = 1 + x*(A(x) + A(x)^3) + x^2*A(x)^4.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x)^2)*x + (1 + 2^2*A(x)^2 + A(x)^4)*x^2/2 +
(1 + 3^2*A(x)^2 + 3^2*A(x)^4 + A(x)^6)*x^3/3 +
(1 + 4^2*A(x)^2 + 6^2*A(x)^4 + 4^2*A(x)^6 + A(x)^8)*x^4/4 +
(1 + 5^2*A(x)^2 + 10^2*A(x)^4 + 10^2*A(x)^6 + 5^2*A(x)^8 + A(x)^10)*x^5/5 +...
more explicitly,
log(A(x)) = 2*x + 14*x^2/2 + 122*x^3/3 + 1118*x^4/4 + 10557*x^5/5 + 101642*x^6/6 + 991916*x^7/7 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A215623, A215624, A215654, A007863, A036765, A198951, A181734, A073157, A364375.

nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=2; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF)*(1 + x*AGF^3) - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 19 2013 *)

a(n):=sum((sum((binomial(2*n+2*k+2,j-k)*binomial(n+2*k,k))/(k+n+1),k,0,j))*(-1)^(n-j)*binomial(2*n-j,n-j),j,0,n); /* Vladimir Kruchinin, Mar 13 2016 */

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(A+x*O(x^n))^(2*j))*x^m/m))); polcoeff(A, n)}

{a(n)=polcoeff((1/x)*serreverse( 2*x^2*(1+x) / (1 - sqrt(1 - 4*x*(1+x)^2 +x^3*O(x^n)))),n)}

{a(n)=local(A=1+x);for(i=1,n,A=(1 + x*A)*(1 + x*(A+x*O(x^n))^3));polcoeff(A,n)}

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(2*k) ).
(2) A(x) = (1/x)*Series_Reversion( 2*x^2*(1+x) / (1 - sqrt(1 - 4*x*(1+x)^2)) ).
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A073157 (Schroeder n-paths containing no FFs).
The formal inverse of g.f. A(x) is (sqrt((1-x^2)^2 + 4*x^3) - (1+x^2)) / (2*x^3).
D-finite with recurrence: 2*n*(n+1)*(2*n+1)*(1275*n^5 - 11696*n^4 + 36827*n^3 - 40618*n^2 - 5828*n + 25368)*a(n) = 6*n*(2*n - 1)*(7650*n^6 - 66351*n^5 + 183953*n^4 - 102147*n^3 - 314787*n^2 + 450754*n - 137760)*a(n-1) - 6*(n-1)*(2*n - 3)*(34425*n^6 - 281367*n^5 + 690471*n^4 - 86579*n^3 - 1831014*n^2 + 2230808*n - 685440)*a(n-2) + 6*(22950*n^8 - 279378*n^7 + 1275447*n^6 - 2461807*n^5 + 518525*n^4 + 5756973*n^3 - 9486182*n^2 + 5962912*n - 1303680)*a(n-3) - 6*(22950*n^8 - 313803*n^7 + 1633059*n^6 - 3736233*n^5 + 1886879*n^4 + 7909228*n^3 - 16107824*n^2 + 11531408*n - 2756544)*a(n-4) + 3*(n-4)*(3*n - 14)*(3*n - 7)*(1275*n^5 - 5321*n^4 + 2793*n^3 + 12437*n^2 - 16992*n + 5328)*a(n-5). - Vaclav Kotesovec, Sep 19 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 10.5255382776611313... is the root of the equation -27 + 108*d - 108*d^2 + 324*d^3 - 72*d^4 + 4*d^5 = 0 and c = 0.5321376859604656812266678970406658537671... - Vaclav Kotesovec, Sep 19 2013
a(n) = Sum_{j=0..n}((Sum_{k=0..j}((binomial(2*n+2*k+2,j-k)*binomial(n+2*k,k))/(k+n+1)))*(-1)^(n-j)*binomial(2*n-j,n-j)). - Vladimir Kruchinin, Mar 13 2016
a(n) = Sum_{k=0..n} binomial(n+2*k+1,k) * binomial(n+2*k+1,n-k) / (n+2*k+1). - Seiichi Manyama, Jul 19 2023

A198888 G.f. A(x) satisfies A(x) = (1 + xA(x))(1 + x^3*A(x)^4).

Original entry on oeis.org

1, 1, 1, 2, 7, 22, 61, 172, 528, 1695, 5447, 17486, 56778, 187064, 622149, 2080325, 6990670, 23621143, 80230388, 273687898, 937072049, 3219316096, 11095261035, 38351414036, 132915860364, 461770505371, 1607875309626, 5610314558562, 19614016834508, 68696001390320, 241007011551493

Offset: 0

Paul D. Hanna, Nov 03 2011

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 61*x^6 + 172*x^7 +...
Related expansions:
A(x)^4 = 1 + 4*x + 10*x^2 + 24*x^3 + 71*x^4 + 236*x^5 + 766*x^6 +...
A(x)^5 = 1 + 5*x + 15*x^2 + 40*x^3 + 120*x^4 + 401*x^5 + 1340*x^6 +...
where A(x) = 1 + x*A(x) + x^3*A(x)^4 + x^4*A(x)^5.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x^2*A(x)^3)*x + (1 + 2^2*x^2*A(x)^3 + x^4*A(x)^6)*x^2/2 +
(1 + 3^2*x^2*A(x)^3 + 3^2*x^4*A(x)^6 + x^6*A(x)^9)*x^3/3 +
(1 + 4^2*x^2*A(x)^3 + 6^2*x^4*A(x)^6 + 4^2*x^6*A(x)^9 + x^8*A(x)^12)*x^4/4 +
(1 + 5^2*x^2*A(x)^3 + 10^2*x^4*A(x)^6 + 10^2*x^6*A(x)^9 + 5^2*x^8*A(x)^12 + x^10*A(x)^15)*x^5/5 +...
Explicitly,
log(A(x)) = x + x^2/2 + 4*x^3/3 + 21*x^4/4 + 76*x^5/5 + 232*x^6/6 + 743*x^7/7 + 2629*x^8/8 + 9481*x^9/9 +...

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A181734, A198957, A198953, A198951, A192415, A036765.

Table[Sum[Binomial[n+k,k]*Binomial[n+k+1,n-3*k]/(n+1),{k,0,Floor[n/3]}],{n,0,20}] (* Vaclav Kotesovec, Sep 18 2013 *)

{a(n)=sum(k=0, n\3, binomial(n+k, k)*binomial(n+k+1, n-3*k))/(n+1)}

{a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^3*(A+x*O(x^n))^4)); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*(x^2*A^3+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, n, binomial(m+j, j)^2*(x^2*A^3+x*O(x^n))^j)*(1-x^2*A^3)^(2*m+1)*x^m/m))); polcoeff(A, n, x)}

a(n) = Sum_{k=0..[n/3]} C(n+k, k)*C(n+k+1, n-3*k)/(n+1).
G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion( x/(1+x) - x^4 ). [Corrected by Seiichi Manyama, Dec 15 2024]
(2) A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1 + x)/(1 - x^3 - x^4).
(3) A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*x^(2*k)*A(x)^(3*k)] * x^n/n ).
(4) A(x) = exp( Sum_{n>=1} [Sum_{k>=0} C(n+k,k)^2*x^(2*k)*A(x)^(3*k)]*(1-x^2*A(x)^3)^(2*n+1)* x^n/n ).
Recurrence: 283*(n-2)*(n-1)*n*(n+1)*(23959952*n^4 - 257205740*n^3 + 1013304652*n^2 - 1735060589*n + 1087154052)*a(n) = 4*(n-2)*(n-1)*n*(8529742912*n^5 - 95830114896*n^4 + 406564828744*n^3 - 799079033082*n^2 + 700270562579*n - 198783157747)*a(n-1) - 8*(n-2)*(n-1)*(8625582720*n^6 - 109845231840*n^5 + 557377471920*n^4 - 1435513153260*n^3 + 1966313576808*n^2 - 1346689501571*n + 355664911636)*a(n-2) + 32*(n-2)*(4216951552*n^7 - 64244492224*n^6 + 407865945256*n^5 - 1396107234938*n^4 + 2774470392903*n^3 - 3187035309382*n^2 + 1946241786026*n - 482103205479)*a(n-3) - 16*(n-2)*(1150077696*n^7 - 19246341696*n^6 + 133834520688*n^5 - 499899483140*n^4 + 1078973257808*n^3 - 1338172075263*n^2 + 875535465587*n - 229801752572)*a(n-4) + 8*(n-4)*(2*n - 5)*(4*n - 17)*(4*n - 11)*(23959952*n^4 - 161365932*n^3 + 385447144*n^2 - 384228697*n + 132152327)*a(n-5). - Vaclav Kotesovec, Sep 18 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 3.686367878047643633... is the root of the equation -256 + 768*d - 5632*d^2 + 2880*d^3 - 1424*d^4 + 283*d^5 = 0 and c = 0.73361916425726935915879240304621641469885... - Vaclav Kotesovec, Sep 18 2013

A199874 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)^2).

Original entry on oeis.org

1, 1, 3, 10, 37, 147, 611, 2625, 11564, 51953, 237123, 1096420, 5125063, 24178427, 114974387, 550511901, 2651896733, 12843003108, 62494595022, 305400429548, 1498184696271, 7375179807191, 36421312544431, 180383163330765, 895756907248150, 4459095182031675, 22247684478181317

Offset: 0

Paul D. Hanna, Nov 11 2011

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 147*x^5 + 611*x^6 +...
where A( x/(1+x^2) - x^2 ) = (1+x^2)/(1-x-x^3).
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 26*x^3 + 103*x^4 + 428*x^5 + 1838*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 80*x^3 + 359*x^4 + 1632*x^5 + 7506*x^6 +...
where A(x) = 1 + x*(1+x)*A(x)^2 + x^3*A(x)^4.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x)*x*A(x) + (1 + 2^2*x + x^2)*x^2*A(x)^2/2 +
(1 + 3^2*x + 3^2*x^2 + x^3)*x^3*A(x)^3/3 +
(1 + 4^2*x + 6^2*x^2 + 4^2*x^3 + x^4)*x^4*A(x)^4/4 +
(1 + 5^2*x + 10^2*x^2 + 10^2*x^3 + 5^2*x^4 + x^5)*x^5*A(x)^5/5 +
(1 + 6^2*x + 15^2*x^2 + 20^2*x^3 + 15^2*x^4 + 6^2*x^5 + x^6)*x^6*A(x)^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 101*x^4/4 + 481*x^5/5 + 2330*x^6/6 +...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.

Cf. A186241, A200074, A200075, A200718, A200719, A215576.

nmax=20;aa=ConstantArray[0,nmax];aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1+x*AGF^2)*(1+x^2*AGF^2)-AGF,x,j]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}];Flatten[{1,aa}] (* Vaclav Kotesovec, Aug 18 2013 *)

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

{a(n)=polcoeff((1/x)*serreverse(x/(1+x^2+x*O(x^n))-x^2),n)}

{a(n)=polcoeff(((1+x^2)/(1-x-x^3+x*O(x^n)))^(n+1)/(n+1), n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j)*x^m*A^m/m))); polcoeff(A, n, x)}

G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion( x/(1+x^2) - x^2 ).
(2) A( x*(1-x-x^3)/(1+x^2) ) = (1+x^2)/(1-x-x^3).
(3) a(n) = [x^n] ((1+x^2)/(1-x-x^3))^(n+1) / (n+1).
(4) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k) * x^n*A(x)^n/n ).
(5) A(x) = exp( Sum_{n>=1} (1-x)^(2*n+1)*(Sum_{k>=0} C(n+k,k)^2*x^k) * x^n*A(x)^n/n ).
Recurrence: 31*(n-1)*n*(n+1)*(85396*n^4 - 902916*n^3 + 3471647*n^2 - 5767203*n + 3503250)*a(n) = 2*(n-1)*n*(6319304*n^5 - 69975436*n^4 + 290875210*n^3 - 559740413*n^2 + 484175751*n - 138985722)*a(n-1) + 2*(n-1)*(2903464*n^6 - 36506072*n^5 + 179801738*n^4 - 439606930*n^3 + 553204983*n^2 - 328951215*n + 67014378)*a(n-2) + 2*(2*n - 5)*(1964108*n^6 - 24695284*n^5 + 123902749*n^4 - 317652203*n^3 + 438313617*n^2 - 307740825*n + 85471038)*a(n-3) - 32*(n-3)*(2*n - 7)*(85396*n^5 - 860218*n^4 + 3249611*n^3 - 5747414*n^2 + 4753791*n - 1471338)*a(n-4) + 8*(n-4)*(n-3)*(2*n - 9)*(85396*n^4 - 561332*n^3 + 1275275*n^2 - 1191073*n + 390174)*a(n-5). - Vaclav Kotesovec, Aug 18 2013
a(n) ~ c*d^n/n^(3/2), where d=5.28245622984... is the root of the equation -16 + 64*d - 92*d^2 - 68*d^3 - 148*d^4 + 31*d^5 = 0 and c = 0.49559010377906722118329... - Vaclav Kotesovec, Aug 18 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-2*k+1,k) * binomial(2*n-2*k+1,n-2*k) / (2*n-2*k+1). - Seiichi Manyama, Jul 18 2023

A192415 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^3*A(x)^2).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 23, 51, 120, 286, 681, 1636, 3985, 9803, 24257, 60338, 150931, 379501, 958360, 2429294, 6179380, 15769380, 40361087, 103579221, 266471500, 687098810, 1775440421, 4596689688, 11922774513, 30977768907, 80615085087, 210103228155, 548352756656, 1433053608502

Offset: 0

Paul D. Hanna, Nov 02 2011

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 23*x^6 + 51*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 15*x^4 + 36*x^5 + 82*x^6 + 190*x^7 +...
A(x)^3 = 1 + 3*x + 6*x^2 + 13*x^3 + 33*x^4 + 84*x^5 + 205*x^6 + 498*x^7 +...
where A(x) = 1 + x*A(x) + x^3*A(x)^2 + x^4*A(x)^3.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x^2*A(x))*x + (1 + 2^2*x^2*A(x) + x^4*A(x)^2)*x^2/2 +
(1 + 3^2*x^2*A(x) + 3^2*x^4*A(x)^2 + x^6*A(x)^3)*x^3/3 +
(1 + 4^2*x^2*A(x) + 6^2*x^4*A(x)^2 + 4^2*x^6*A(x)^3 + x^8*A(x)^4)*x^4/4 +
(1 + 5^2*x^2*A(x) + 10^2*x^4*A(x)^2 + 10^2*x^6*A(x)^3 + 5^2*x^8*A(x)^4 + x^10*A(x)^5)*x^5/5 +...
Explicitly,
log(A(x)) = x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 31*x^5/5 + 70*x^6/6 + 176*x^7/7 + 469*x^8/8 + 1228*x^9/9 + 3161*x^10/10 +...

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A198951, A198953, A198957, A007863, A036765, A104545.

nmax=20; aa=ConstantArray[0,nmax]; aa[[1]]=1; Do[AGF=1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[Coefficient[(1 + x*AGF)*(1 + x^3*AGF^2) - AGF,x,j]==0,koef][[1]];aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}] (* Vaclav Kotesovec, Sep 19 2013 *)

{a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^3*(A+x*O(x^n))^2)); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^(2*j)*(A+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

x='x; y='y; Fxy = (1+x*y) * (1 + x^3*y^2) - y;
seq(N) = {
  my(y0 = 1 + O('x^N), y1=0);
  for (k = 1, N,
    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy,y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
seq(34)  \\ Gheorghe Coserea, Nov 30 2016

G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^(2*k) * A(x)^k ).
D-finite with recurrence: 4*(n+1)*(n+2)*(217*n^3 - 1239*n^2 + 1838*n - 336)*a(n) = 6*(n+1)*(434*n^4 - 2261*n^3 + 2339*n^2 + 1792*n - 1344)*a(n-1) - (n-1)*(2821*n^4 - 13286*n^3 + 7829*n^2 + 18464*n - 4032)*a(n-2) + 6*(868*n^5 - 6258*n^4 + 13981*n^3 - 7438*n^2 - 7769*n + 6136)*a(n-3) + 2*(n-3)*(2*n - 5)*(434*n^3 - 1393*n^2 + 211*n + 1048)*a(n-4) + 2*(n-4)*(2*n - 7)*(217*n^3 - 588*n^2 + 11*n + 480)*a(n-5). - Vaclav Kotesovec, Sep 19 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 2.730683387097269698... is the root of the equation -4 - 8*d - 24*d^2 + 13*d^3 - 12*d^4 + 4*d^5 = 0 and c = 2.078548317061344694159945441842754... is the root of the equation -1 - 67*c^2 - 19811*c^4 + 36463*c^6 - 41664*c^8 + 7936*c^10 = 0. - Vaclav Kotesovec, Sep 19 2013, updated Nov 28 2016
a(n) = Sum_{k=0..n/2+1} C(n-k+2,k-1)*C(n-k+2,2*k-1)/(n-k+2). - Vladimir Kruchinin, Feb 12 2019

A198957 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^2*A(x)^4).

Original entry on oeis.org

1, 1, 2, 7, 26, 102, 424, 1827, 8078, 36466, 167376, 778718, 3664164, 17407068, 83375616, 402198915, 1952296598, 9528757098, 46735576816, 230227356906, 1138609205372, 5651170500612, 28138939936704, 140527262919342, 703704207921932, 3532664478314484, 17775185122527776

Offset: 0

Paul D. Hanna, Nov 01 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 102*x^5 + 424*x^6 + 1827*x^7 +...
Related expansions:
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 237*x^4 + 1028*x^5 + 4570*x^6 +...
A(x)^5 = 1 + 5*x + 20*x^2 + 85*x^3 + 375*x^4 + 1681*x^5 + 7660*x^6 +...
where A(x) = 1 + x*A(x) + x^2*A(x)^4 + x^3*A(x)^5.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A(x)^3)*x + (1 + 2^2*x*A(x)^3 + x^2*A(x)^6)*x^2/2 +
(1 + 3^2*x*A(x)^3 + 3^2*x^2*A(x)^6 + x^3*A(x)^9)*x^3/3 +
(1 + 4^2*x*A(x)^3 + 6^2*x^2*A(x)^6 + 4^2*x^3*A(x)^9 + x^4*A(x)^12)*x^4/4 +
(1 + 5^2*x*A(x)^3 + 10^2*x^2*A(x)^6 + 10^2*x^3*A(x)^9 + 5^2*x^4*A(x)^12 + x^5*A(x)^15)*x^5/5 +...
more explicitly,
log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 75*x^4/4 + 356*x^5/5 + 1746*x^6/6 + 8660*x^7/7 + 43299*x^8/8 +...
Also, g.f. A(x) = G(x*A(x)) where G(x) = A(x/G(x)) (g.f. of A104545) begins:
G(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 55*x^7 + 129*x^8 +...

Cf. A198953, A198951, A007863, A036765, A104545.

CoefficientList[1/x*InverseSeries[Series[2*x^3*(1+x)/(1 - Sqrt[1-4*x^2*(1+x)^2]), {x, 0, 20}], x],x] (* Vaclav Kotesovec, May 28 2014 *)

a(n):=sum(binomial(2*j+n,j)*binomial(2*j+n+1,4*j+1)/(n+j+1),j,0,(n)/2); /* Vladimir Kruchinin, May 28 2014 */

{a(n)=local(A=1+x); for(i=1, n, A=(1 + x*A)*(1 + x^2*(A+x*O(x^n))^4)); polcoeff(A, n)}

{a(n)=polcoeff((1/x)*serreverse( 2*x^3*(1+x)/(1 - sqrt(1-4*x^2*(1+x +x^3*O(x^n))^2))), n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j*(A^3+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x*A^3)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A^3+x*O(x^n))^j)*x^m/m))); polcoeff(A, n, x)}

x='x; y='y; Fxy = (1 + x*y)*(1 + x^2*y^4) - y;
seq(N) = {
  my(y0 = 1 + O('x^N), y1=0);
  for (k = 1, N,
    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
seq(27) \\ Gheorghe Coserea, Nov 30 2016

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(3*k) ).
(2) A(x) = (1/x)*Series_Reversion( 2*x^3*(1+x)/(1 - sqrt(1-4*x^2*(1+x)^2)) ).
(3) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A104545 (Motzkin paths of length n having no consecutive (1,0) steps).
(4) A(x) = exp( Sum_{n>=1} x^n/n * (1-x*A(x)^3)^(2*n+1)*Sum_{k>=0} C(n+k,k)^2 * x^k * A(x)^(3*k) ).
a(n) = sum(j=0..n/2, binomial(2*j+n,j)*binomial(2*j+n+1,4*j+1)/(n+j+1)). - Vladimir Kruchinin, May 28 2014
a(n) ~ sqrt((1 + 2*r*s^3 + 3*r^2*s^4)/(2*Pi*s*(3 + 5*r*s))) / (2*n^(3/2)*r^(n+1/2)), where r = 0.187614989725738719..., s = 1.61178302212918247... are roots of the system of equations r + 4*r^2*s^3 + 5*r^3*s^4 = 1, (1+r*s)*(1+r^2*s^4) = s. - Vaclav Kotesovec, May 28 2014

A200718 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x^2*A(x)^6).

Original entry on oeis.org

1, 1, 3, 14, 75, 433, 2636, 16668, 108399, 720431, 4871555, 33409042, 231817448, 1624503716, 11480658056, 81731416480, 585579734959, 4219179476875, 30552067317233, 222225174139730, 1622894404239115, 11894991079960721, 87472260252499560, 645183802300787356, 4771926560361458884

Offset: 0

Paul D. Hanna, Nov 21 2011

nonn

More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)).

			G.f.: A(x) = 1 + x + 3*x^2 + 14*x^3 + 75*x^4 + 433*x^5 + 2636*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 34*x^3 + 187*x^4 + 1100*x^5 + 6784*x^6 +...
A(x)^6 = 1 + 6*x + 33*x^2 + 194*x^3 + 1200*x^4 + 7674*x^5 + 50317*x^6 +...
A(x)^8 = 1 + 8*x + 52*x^2 + 336*x^3 + 2210*x^4 + 14776*x^5 + 100216*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x)^6 + x^3*A(x)^8.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A^4)*x*A + (1 + 2^2*x*A^4 + x^2*A^8)*x^2*A^2/2 +
(1 + 3^2*x*A^4 + 3^2*x^2*A^8 + x^3*A^12)*x^3*A^3/3 +
(1 + 4^2*x*A^4 + 6^2*x^2*A^8 + 4^2*x^3*A^12 + x^4*A^16)*x^4*A^4/4 +
(1 + 5^2*x*A^4 + 10^2*x^2*A^8 + 10^2*x^3*A^12 + 5^2*x^4*A^16 + x^5*A^20)*x^5*A^5/5 + ...
The g.f. of A104545, G(x) = A(x/G(x)^2) where A(x) = G(x*A(x)^2), begins:
G(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 55*x^7 + 129*x^8 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Recurrence (of order 6)

Cf. A104545, A200716, A200717, A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.

Cf. A186241, A199874, A200074, A200075, A200719, A215576.

a[n_] := Sum[Binomial[2*n + 2*k + 1, k]*Binomial[2*n + 2*k + 1, n - 2*k]/ (2*n + 2*k + 1), {k, 0, n/2}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jan 09 2018, after Vladimir Kruchinin *)

a(n):=sum((binomial(2*n+2*k+1,k)*binomial(2*n+2*k+1,n-2*k))/(2*n+2*k+1),k,0,(n)/2); /* Vladimir Kruchinin, Mar 11 2016 */

{a(n)=polcoeff(sqrt( (1/x)*serreverse( 2*x^5*(1+x)^2/(1 - 2*x^2*(1+x)^2 - sqrt(1 - 4*x^2*(1+x)^2+O(x^(n+6)))) ) ),n)}

{a(n)=local(p=1,q=4,A=1+x);for(i=1,n,A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n));polcoeff(A,n)}

{a(n)=local(p=1,q=4,A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0,m,binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

{a(n)=local(p=1,q=4,A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

G.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion( 2*x^5*(1+x)^2/(1 - 2*x^2*(1+x)^2 - sqrt(1 - 4*x^2*(1+x)^2)) ) ).
(2) A(x) = G(x*A(x)^2) where G(x) = A(x/G(x)^2) is the g.f. of A104545 (Motzkin paths of length n having no consecutive (1,0) steps).
(3) A(x) = exp( Sum_{n>=1} x^n * A(x)^n/n * [Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(4*k)] ).
(4) A(x) = exp( Sum_{n>=1} x^n * A(x)^n/n * [(1-x/A(x)^4)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2*x^k * A(x)^(4*k)] ).
a(n) = Sum_{k=0..floor(n/2)}((binomial(2*n+2*k+1,k)*binomial(2*n+2*k+1,n-2*k))/(2*n+2*k+1)). - Vladimir Kruchinin, Mar 11 2016

A200719 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x^2*A(x)^5).

Original entry on oeis.org

1, 1, 3, 13, 64, 340, 1903, 11053, 65993, 402527, 2497439, 15712220, 100001459, 642719263, 4165537744, 27193644061, 178654643151, 1180282875483, 7836312619243, 52259258911091, 349902441457427, 2351240866736891, 15851508780927739, 107187240225220684, 726784821098903319

Offset: 0

Paul D. Hanna, Nov 21 2011

nonn

More generally, for fixed parameters p and q, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^n * F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^k * F(x)^(k*q)] ),
then F(x) = (1 + x*F(x)^(p+1))*(1 + x^2*F(x)^(p+q+1)).

			G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 64*x^4 + 340*x^5 + 1903*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 32*x^3 + 163*x^4 + 886*x^5 + 5039*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 135*x^3 + 765*x^4 + 4481*x^5 + 26920*x^6 +...
A(x)^7 = 1 + 7*x + 42*x^2 + 252*x^3 + 1533*x^4 + 9457*x^5 + 59101*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x)^5 + x^3*A(x)^7.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A^3)*x*A + (1 + 2^2*x*A^3 + x^2*A^6)*x^2*A^2/2 +
(1 + 3^2*x*A^3 + 3^2*x^2*A^6 + x^3*A^9)*x^3*A^3/3 +
(1 + 4^2*x*A^3 + 6^2*x^2*A^6 + 4^2*x^3*A^9 + x^4*A^12)*x^4*A^4/4 +
(1 + 5^2*x*A^3 + 10^2*x^2*A^6 + 10^2*x^3*A^9 + 5^2*x^4*A^12 + x^5*A^15)*x^5*A^5/5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..300 from Vaclav Kotesovec)

Cf. A200716, A200717, A200718, A200074, A200075, A199876, A199874, A199877, A198951, A198953, A198957, A192415, A198888, A036765, A104545.

Cf. A186241, A199874, A200074, A200075, A200718, A215576.

{a(n)=local(p=1,q=3,A=1+x);for(i=1,n,A=(1+x*A^(p+1))*(1+x^2*A^(p+q+1))+x*O(x^n));polcoeff(A,n)}

{a(n)=local(p=1,q=3,A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m*(A+x*O(x^n))^(p*m)/m*sum(j=0,m,binomial(m, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

{a(n)=local(p=1,q=3,A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m*(A+x*O(x^n))^(p*m)/m*(1-x*A^q)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*(A+x*O(x^n))^(q*j))))); polcoeff(A, n, x)}

G.f. A(x) satisfies:
(1) A(x) = exp( Sum_{n>=1} x^n * A(x)^n/n * [Sum_{k=0..n} C(n,k)^2 * x^k * A(x)^(3*k)] ).
(2) A(x) = exp( Sum_{n>=1} x^n * A(x)^n/n * [(1-x/A(x)^3)^(2*n+1) * Sum_{k>=0} C(n+k,k)^2*x^k * A(x)^(3*k)] ).
Recurrence: 4232*(n-2)*(n-1)*n*(2*n - 3)*(2*n - 1)*(2*n + 1)*(108983978975*n^7 - 1828734495225*n^6 + 13017379495661*n^5 - 50928975062019*n^4 + 118201965098732*n^3 - 162617590602876*n^2 + 122676758610192*n - 39103265134080)*a(n) = 8*(n-2)*(n-1)*(2*n - 3)*(2*n - 1)*(850837923857825*n^9 - 15127768128079400*n^8 + 116088908648008427*n^7 - 502364025369222635*n^6 + 1342887860190877280*n^5 - 2280899268898038065*n^4 + 2433907848768834828*n^3 - 1548429898790214180*n^2 + 521138603722292640*n - 68863424146977600)*a(n-1) - 30*(n-2)*(2*n - 3)*(155302170039375*n^11 - 3227155335853125*n^10 + 29807524885054600*n^9 - 161278340404759950*n^8 + 566950865855228019*n^7 - 1356848300481904461*n^6 + 2250482361655315470*n^5 - 2579665279074165840*n^4 + 1996011605601581864*n^3 - 988803599084885136*n^2 + 280851990522009984*n - 34444332223983360)*a(n-2) + 5*(7288303593953125*n^13 - 187891351713750000*n^12 + 2204843674914291875*n^11 - 15579013461781304250*n^10 + 73867718896175411475*n^9 - 247858726321141236540*n^8 + 604530296941440837821*n^7 - 1082990060568950070282*n^6 + 1421457900098213642392*n^5 - 1345695224728829837040*n^4 + 889319601933492222864*n^3 - 386196670582228097568*n^2 + 97916706472751405568*n - 10797892365692920320)*a(n-3) + 10*(n-3)*(2*n - 7)*(5*n - 18)*(5*n - 14)*(5*n - 12)*(5*n - 11)*(108983978975*n^7 - 1065846642400*n^6 + 4333636082786*n^5 - 9458655747964*n^4 + 11899609166891*n^3 - 8554104592084*n^2 + 3208950812340*n - 473478110640)*a(n-4). - Vaclav Kotesovec, Nov 17 2017
a(n) ~ s * sqrt((r*s*(r*s^3 - 1) - 3) / (7*Pi*(5*r*s*(1 + r*s^3) - 3))) / (2*n^(3/2)*r^n), where r = 0.1385102270697349252376651829944449360743895474888... and s = 1.450646440303399446510765649245639306003224666768... are real roots of the system of equations (1 + r*s^2)*(1 + r^2*s^5) = s, r*s*(2 + 5*r*s^3 + 7*r^2*s^5) = 1. - Vaclav Kotesovec, Nov 22 2017
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(2*n+k+1,n-2*k) / (2*n+k+1). - Seiichi Manyama, Jul 18 2023

A215654 G.f. satisfies: A(x) = (1 + xA(x)^2) (1 + x*A(x)^3).

Original entry on oeis.org

1, 2, 11, 81, 684, 6257, 60325, 603641, 6210059, 65272503, 697898849, 7566847547, 82999675563, 919376968734, 10269588489433, 115548651723889, 1308374198000780, 14897993185500455, 170482798370871370, 1959574731164246402, 22614008012647634411, 261915716386286916342

Offset: 0

Paul D. Hanna, Aug 19 2012

nonn

More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = (1 + x^r*F(x)^(p+1)) * (1 + x^(r+s)*F(x)^(p+q+1)), then
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ).
The radius of convergence of g.f. A(x) is r = 0.08035832347291483065438962031... with A(r) = 1.5393913914574609282262181402132760790902539070... where y=A(r) satisfies 20*y^3 - 38*y^2 + 15*y - 6 = 0.
r = 1/(187/300*17^(2/3) + 119/75*17^(1/3) + 1273/300). - Vaclav Kotesovec, Sep 17 2013
Number of hybrid ternary trees with n internal nodes. [Hong and Park]. - N. J. A. Sloane, Mar 26 2014

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 81*x^3 + 684*x^4 + 6257*x^5 + 60325*x^6 +...
Related expansions.
A(x)^2 = 1 + 4*x + 26*x^2 + 206*x^3 + 1813*x^4 + 17032*x^5 +...
A(x)^3 = 1 + 6*x + 45*x^2 + 383*x^3 + 3519*x^4 + 34023*x^5 +...
A(x)^5 = 1 + 10*x + 95*x^2 + 925*x^3 + 9270*x^4 + 95237*x^5 +...
where A(x) = 1 + x*(A(x)^2 + A(x)^3) + x^2*A(x)^5.
The g.f. also satisfies the series:
A(x) = 1 + 2*x*A(x)^2 + 3*x^2*A(x)^4 + 5*x^3*A(x)^6 + 8*x^4*A(x)^8 + 13*x^5*A(x)^10 + 21*x^6*A(x)^12 + 34*x^7*A(x)^14 +...+ Fibonacci(n+2)*x^n*A(x)^(2*n) +...
and consequently, A( x*(1-x-x^2)^2/(1+x)^2 ) = (1+x)/(1-x-x^2).
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x))*x*A(x) + (1 + 2^2*A(x) + A(x)^2)*x^2*A(x)^2/2 +
(1 + 3^2*A(x) + 3^2*A(x)^2 + A(x)^3)*x^3*A(x)^3/3 +
(1 + 4^2*A(x) + 6^2*A(x)^2 + 4^2*A(x)^3 + A(x)^4)*x^4*A(x)^4/4 +
(1 + 5^2*A(x) + 10^2*A(x)^2 + 10^2*A(x)^3 + 5^2*A(x)^4 + A(x)^5)*x^5*A(x)^5/5 +...
Explicitly,
log(A(x)) = 2*x + 18*x^2/2 + 185*x^3/3 + 2006*x^4/4 + 22412*x^5/5 + 255249*x^6/6 + 2946155*x^7/7 + 34342270*x^8/8 +...+ L(n)*x^n/n +...
where L(n) = [x^n] (1+x)^(2*n)/(1-x-x^2)^(2*n) / 2.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
SeoungJi Hong and SeungKyung Park, Hybrid d-ary trees and their generalization, Bull. Korean Math. Soc. 51 (2014), No. 1, pp. 229-235. See p. 233. - _N. J. A. Sloane_, Mar 26 2014
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Cf. A007863, A198953, A215624, A198951.

Column k=3 of A245049.

a:= n-> coeff(series(RootOf((1+x*A^2)*(1+x*A^3)-A, A), x, n+1), x, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Apr 04 2019

CoefficientList[Sqrt[1/x*InverseSeries[Series[x*(1-x-x^2)^2/(1+x)^2,{x,0,20}],x]],x] (* Vaclav Kotesovec, Sep 17 2013 *)

a(n):=sum(binomial(2*n+i,i)*binomial(2*n+i+1,n-i),i,0,n)/(2*n+1); /* Vladimir Kruchinin, Apr 04 2019 */

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1 + x*A^2)*(1 + x*A^3)); polcoeff(A, n)}

{a(n)=polcoeff(sqrt((1/x)*serreverse( x*(1-x-x^2)^2/(1+x +x*O(x^n))^2)), n)}
for(n=0,31,print1(a(n),", "))

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*A^j)*x^m*A^m/m))); polcoeff(A, n)}

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2/A^j)*x^m*A^(2*m)/m))); polcoeff(A, n)}

{a(n)=polcoeff(((1+x)/(1-x-x^2 +x*O(x^n)))^(2*n+1)/(2*n+1),n)}

G.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion( x*(1-x-x^2)^2/(1+x)^2 ) ).
(2) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^k ).
(3) A(x) = exp( Sum_{n>=1} x^n*A(x)^(2*n)/n * Sum_{k=0..n} C(n,k)^2 / A(x)^k ).
(4) A(x) = Sum_{n>=0} Fibonacci(n+2) * x^n * A(x)^(2*n).
(5) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A007863 (number of hybrid binary trees with n internal nodes).
The formal inverse of g.f. A(x) is (sqrt(1-2*x+5*x^2) - (1+x))/(2*x^3).
a(n) = [x^n] ( (1+x)/(1-x-x^2) )^(2*n+1) / (2*n+1).
Recurrence: 100*(n-1)*n*(2*n-1)*(2*n+1)*(4913*n^3 - 26877*n^2 + 49912*n - 30480)*a(n) = 2*(n-1)*(2*n-1)*(6254249*n^5 - 40468670*n^4 + 99110119*n^3 - 109861414*n^2 + 52822608*n - 8566560)*a(n-1) - 3*(2343501*n^7 - 22194333*n^6 + 87905623*n^5 - 187987155*n^4 + 233161624*n^3 - 166253172*n^2 + 62010112*n - 8952000)*a(n-2) + 6*(n-2)*(2*n-5)*(3*n-8)*(3*n-4)*(4913*n^3 - 12138*n^2 + 10897*n - 2532)*a(n-3). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ 1/1020*sqrt(73695 + 11730*17^(2/3) + 28815*17^(1/3)) * (187/300*17^(2/3) + 119/75*17^(1/3) + 1273/300)^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013
a(n) = 1/(2*n+1)*Sum_{i=0..n} C(2*n+i,i)*C(2*n+i+1,n-i). - Vladimir Kruchinin, Apr 04 2019

A200074 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)).

Original entry on oeis.org

1, 1, 3, 9, 30, 108, 406, 1577, 6280, 25499, 105169, 439388, 1855636, 7908909, 33975250, 146954693, 639460707, 2797384235, 12295494109, 54272825103, 240480529815, 1069257987503, 4769306203838, 21334400243252, 95687482105807, 430217846136134, 1938651904470374, 8754225470415889

Offset: 0

Paul D. Hanna, Nov 13 2011

nonn

More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ),
then F(x) = (1 + x^r*F(x)^(p+1))*(1 + x^(r+s)*F(x)^(p+q+1)).

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 108*x^5 + 406*x^6 + 1577*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 24*x^3 + 87*x^4 + 330*x^5 + 1289*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 46*x^3 + 180*x^4 + 720*x^5 + 2928*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x) + x^3*A(x)^3.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (A + x)*x + (A^2 + 2^2*x*A + x^2)*x^2/2 +
(A^3 + 3^2*x*A^2 + 3^2*x^2*A + x^3)*x^3/3 +
(A^4 + 4^2*x*A^3 + 6^2*x^2*A^2 + 4^2*x^3*A + x^4)*x^4/4 +
(A^5 + 5^2*x*A^4 + 10^2*x^2*A^3 + 10^2*x^3*A^2 + 5^2*x^4*A + x^5)*x^5/5 +
(A^6 + 6^2*x*A^5 + 15^2*x^2*A^4 + 20^2*x^3*A^3 + 15^2*x^4*A^2 + 6^2*x^5*A + x^6)*x^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 77*x^4/4 + 331*x^5/5 + 1445*x^6/6 + 6392*x^7/7 + 28565*x^8/8 +...

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A199876, A199877, A198951, A198953, A198957, A192415, A198888, A004148, A036765.

Cf. A186241, A199874, A200075, A200718, A200719, A215576.

a:= n-> coeff(series(RootOf(A=(1+x*A^2)*(1+x^2*A), A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2012

m = 28; A[_] = 0;
Do[A[x_] = (1 + x A[x]^2)(1 + x^2 A[x]) + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 02 2019 *)

{a(n)=local(A=1+x);for(i=1,n,A=(1+x*A^2)*(1+x^2*A^1)+x*O(x^n));polcoeff(A,n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j/A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x/A)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j/A^j)*x^m*A^m/m))); polcoeff(A, n, x)}

G.f. satisfies:
(1) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^(n-k)) * x^n/n ).
(2) A(x) = exp( Sum_{n>=1} (1-x/A(x))^(2*n+1)*(Sum_{k>=0} C(n+k,k)^2*x^k/A(x)^k) * x^n*A(x)^n/n ).
(3) A(x) = x / Series_Reversion( x*G(x) ) where G(x) is the g.f. of A199876.
(4) A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A199876.
Recurrence: (n+1)*(n+2)*(1241*n^4 - 10636*n^3 + 25417*n^2 - 7382*n - 17136)*a(n) =  - 18*(n+1)*(443*n^3 - 3889*n^2 + 9734*n - 5712)*a(n-1) + 4*(6205*n^6 - 53180*n^5 + 115741*n^4 + 64762*n^3 - 370103*n^2 + 246727*n - 25704)*a(n-2) + 6*(2482*n^6 - 24995*n^5 + 76519*n^4 - 36347*n^3 - 185471*n^2 + 293092*n - 140400)*a(n-3) + 2*(4964*n^6 - 57436*n^5 + 228617*n^4 - 276802*n^3 - 361447*n^2 + 956696*n - 320496)*a(n-4) - 6*(2482*n^6 - 32441*n^5 + 140587*n^4 - 173153*n^3 - 266705*n^2 + 677518*n - 291840)*a(n-5) + 12*(n-4)*(2*n - 11)*(11*n^2 + 73*n - 748)*a(n-6) + 2*(n-5)*(2*n - 13)*(1241*n^4 - 5672*n^3 + 955*n^2 + 16508*n - 8496)*a(n-7). - Vaclav Kotesovec, Aug 18 2013
a(n) ~ c*d^n/n^(3/2), where d = 4.770539985405... is the root of the equation -4 + 12*d^2 - 8*d^3 - 12*d^4 - 20*d^5 + d^7 = 0 and c = 0.612892860188927397373456... - Vaclav Kotesovec, Aug 18 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k+1,k) * binomial(2*n-3*k+1,n-2*k) / (2*n-3*k+1). - Seiichi Manyama, Jul 18 2023

cp's OEIS Frontend

A036765 Number of ordered rooted trees with n non-root nodes and all outdegrees <= three.

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A198953 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^3).

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A198888 G.f. A(x) satisfies A(x) = (1 + x*A(x))*(1 + x^3*A(x)^4).

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A199874 G.f. satisfies A(x) = (1 + x*A(x)^2)*(1 + x^2*A(x)^2).

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A192415 G.f. satisfies: A(x) = (1 + x*A(x))*(1 + x^3*A(x)^2).

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A198957 G.f. satisfies: A(x) = (1 + x*A(x))*(1 + x^2*A(x)^4).

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A198953 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^3).

A198888 G.f. A(x) satisfies A(x) = (1 + xA(x))(1 + x^3*A(x)^4).

A199874 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)^2).

A192415 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^3*A(x)^2).

A198957 G.f. satisfies: A(x) = (1 + xA(x))(1 + x^2*A(x)^4).

A200718 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x^2*A(x)^6).

A200719 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x^2*A(x)^5).

A215654 G.f. satisfies: A(x) = (1 + xA(x)^2) (1 + x*A(x)^3).

A200074 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)).