A036765 -id:A036765 - cp's OEIS frontend

A369440 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x^2)^2) ).

1, 1, 3, 9, 30, 107, 396, 1513, 5915, 23554, 95202, 389555, 1610588, 6717816, 28234064, 119452553, 508330809, 2174393331, 9343913933, 40319400738, 174630125428, 758916134002, 3308320668768, 14462616815619, 63388694309005, 278492994845776, 1226241871745376

Offset: 0

Seiichi Manyama, Jan 23 2024

nonn

Index entries for reversions of series

Cf. A036765, A369441.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+x^2)^2))/x)

a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial(u*(n+1), n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+2,k) * binomial(n+1,n-2*k).

A002844 Number of non-isentropic binary rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 13, 36, 102, 296, 871, 2599, 7830, 23799, 72855, 224455, 695303, 2164491

Offset: 1

N. J. A. Sloane

From Richard Guy's 1971 letter: "[Studied by] Helen Alderson, J. H. Conway, etc. at Cambridge. These are rooted trees with two branches at each stage and if A,B,C,D (see drawing [in letter]) are further growths, then one treats (AB)(CD) as equivalent to (AC)(BD) - otherwise one distinguishes left and right. [The sequence gives] the number of equivalence classes of such trees."

R. K. Guy, personal communication.
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).

Tyler Foster, A Noncommutative Version of the Natural Numbers, arXiv:1003.2081 [math.QA], 2010. See D(n) Table 2 p. 3.
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission] See sequence C.
N. J. A. Sloane, Winter Fruits: New Problems from the OEIS, Dec. 2016 - Jan. 2017 (Part 1), Jan 26 2017.
N. J. A. Sloane, Winter Fruits: New Problems from the OEIS, Dec. 2016 - Jan. 2017 (slides)
Doron Zeilberger, Maple program for A002844
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Bears a superficial resemblance to A036765.

Revised by N. J. A. Sloane, Dec 15 2016
a(11)-a(14) from Doron Zeilberger, Jan 31 2017
a(15)-a(16) from Sean A. Irvine, Sep 29 2023

A135305 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UUUU's.

Original entry on oeis.org

1, 1, 2, 5, 13, 1, 36, 5, 1, 104, 21, 6, 1, 309, 84, 28, 7, 1, 939, 322, 124, 36, 8, 1, 2905, 1206, 522, 174, 45, 9, 1, 9118, 4455, 2127, 795, 235, 55, 10, 1, 28964, 16302, 8492, 3487, 1155, 308, 66, 11, 1, 92940, 59268, 33396, 14894, 5412, 1617, 394, 78, 12, 1

Offset: 0

N. J. A. Sloane, Dec 07 2007

Each of rows 0, 1, 2, 3 has one entry. Row n (n >= 3) has n-2 entries. Row sums yield the Catalan numbers (A000108). Column 0 yields A036765. - Emeric Deutsch, Dec 14 2007

			Triangle begins:
1
1
2
5
13 1
36 5 1
104 21 6 1
309 84 28 7 1
...
T(5,1) = 5 because we have UUUUDUDDDD, UUUUDDUDDD, UUUUDDDUDD, UUUUDDDDUD and UDUUUUDDDD.

Alois P. Heinz, Rows n = 0..150, flattened
FindStat - Combinatorial Statistic Finder, The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]].
A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924.

Cf. A000108, A036765, A243752.

eq:=(1-t)*z^3*G^3+z*(t+z-t*z)*G^2+((1-t)*z-1)*G+1: g:=RootOf(eq,G): gser:= simplify(series(g,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(gser,z,n)) end do: 1;1;2; for n from 3 to 12 do seq(coeff(P[n],t,j),j=0..n-3) end do; # yields sequence in triangular form; # Emeric Deutsch, Dec 14 2007
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
      `if`(x=0, 1, expand(b(x-1, y+1, min(t+1, 4))*
      `if`(t=4, z, 1) +b(x-1, y-1, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..15);  # Alois P. Heinz, Jun 02 2014

b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, Min[t+1, 4]]*If[t == 4, z, 1] + b[x-1, y-1, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]] @ b[2*n, 0, 1]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Nov 28 2014, after Alois P. Heinz *)

G.f.: G=G(t,z) satisfies (1-t)*z^3*G^3 + z*(t+z-t*z)*G^2 + ((1-t)*z-1)*G+1 = 0. - Emeric Deutsch, Dec 14 2007

Edited and extended by Emeric Deutsch, Dec 14 2007

A198944 G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^kA(x)^k] x^n/n ).

Original entry on oeis.org

1, 1, 2, 7, 23, 78, 291, 1126, 4436, 17910, 73773, 308188, 1303402, 5573133, 24050795, 104620985, 458324429, 2020417339, 8956142180, 39899217350, 178549985024, 802275736073, 3618237414959, 16373514195570, 74325340129430, 338356926399193, 1544406450870590

Offset: 0

Paul D. Hanna, Oct 31 2011

nonn

Compare to a g.f. G(x) of A036765 (rooted trees with a degree constraint):

G(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2 * x^k*G(x)^k] * x^n/n ).

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 23*x^4 + 78*x^5 + 291*x^6 +...
where
log(A(x)) = (1 + x*A(x))*x + (1 + 2^3*x*A(x) + x^2*A(x)^2)*x^2/2 +
(1 + 3^3*x*A(x) + 3^3*x^2*A(x)^2 + x^3*A(x)^3)*x^3/3 +
(1 + 4^3*x*A(x) + 6^3*x^2*A(x)^2 + 4^3*x^3*A(x)^3 + x^4*A(x)^4)*x^4/4 +
(1 + 5^3*x*A(x) + 10^3*x^2*A(x)^2 + 10^3*x^3*A(x)^3 + 5^3*x^4*A(x)^4 + x^5*A(x)^5)*x^5/5 +...
more explicitly,
log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 63*x^4/4 + 251*x^5/5 + 1110*x^6/6 +...

Cf. A036765, A192131.

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

A303271 Number of ordered rooted trees with n non-root nodes such that the maximal outdegree equals three.

Original entry on oeis.org

1, 4, 15, 53, 182, 616, 2070, 6930, 23166, 77429, 258973, 867230, 2908633, 9772556, 32896088, 110949072, 374934201, 1269505482, 4306750577, 14638006449, 49843505965, 170021694271, 580954640775, 1988357053020, 6816047416230, 23400699072231, 80455436055699

Offset: 3

Alois P. Heinz, Apr 20 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..1802

Column k=3 of A203717.

Cf. A001006, A036765.

b:= proc(u, o, k) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, k), j=1..min(1, u))+
      add(b(u+j-1, o-j, k), j=1..min(k, o)))
    end:
a:= n-> b(0, n, 3)-b(0, n, 2):
seq(a(n), n=3..35);

a(n) = A036765(n) - A001006(n).

A337512 G.f. A(x) satisfies: A(x) = 1 - Sum_{k=1..3} (x * A(x))^k.

Original entry on oeis.org

1, -1, 0, 1, 1, -6, 4, 13, -13, -61, 124, 120, -516, -352, 2848, -923, -11337, 10165, 49352, -88655, -159903, 512430, 450812, -2873276, -11660, 13752804, -9160464, -62238760, 91526344, 239932224, -620180224, -768156379, 3683079807, 1168683353

Offset: 0

Ilya Gutkovskiy, Aug 30 2020

sign

Cf. A000073, A007440, A036765, A337513, A337514.

nmax = 33; A[] = 0; Do[A[x] = 1 - Sum[(x A[x])^k, {k, 1, 3}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 34; CoefficientList[(1/x) InverseSeries[Series[x/(1 - x - x^2 - x^3), {x, 0, nmax}], x], x]
b[m_, r_, k_] := b[m, r, k] = If[m + r == 0, 1, Sum[b[m - j, r + j - 1, k], {j, 1, Min[1, m]}] - Sum[b[m + j - 1, r - j, k], {j, 1, Min[k, r]}]]; a[n_] := b[0, n, 3]; Table[a[n], {n, 0, 33}]

G.f.: A(x) = (1/x) * Series_Reversion(x / (1 - x - x^2 - x^3)).

A378425 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^2 * (1 + x)^3) ).

Original entry on oeis.org

1, 1, 2, 7, 24, 82, 297, 1121, 4317, 16900, 67185, 270480, 1100122, 4513809, 18661618, 77666327, 325117967, 1368001765, 5782686120, 24545144206, 104573104040, 447036252525, 1916918691196, 8243075111450, 35538551601880, 153584392913986, 665201585797986, 2887012910233897

Offset: 0

Seiichi Manyama, Nov 25 2024

nonn

Index entries for reversions of series

Cf. A036765, A378426.
Cf. A300048, A378427.
Cf. A378406.

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1+x+x^2*(1+x)^3))/x)

a(n) = sum(k=0, n\2, binomial(n+1, k)*binomial(n+2*k+1, n-2*k))/(n+1);

G.f.: exp( Sum_{k>=1} A378406(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] (1 + x + x^2 * (1 + x)^3)^(n+1).
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(n+1,k) * binomial(n+2*k+1,n-2*k).

A059967 Number of 9-ary trees.

Original entry on oeis.org

1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, 1416298046436, 28748759731965, 589546754316126, 12195537924351375, 254184908607118800, 5332692942907262361, 112524941404978156215

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Mar 05 2001

nonn

S. Heubach, N. Y. Li and T. Mansour, Staircase tilings and k-Catalan structures, Discrete Math., 308 (2008), 5954-5964.
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

Related algebraic sequences concerning trees: strictly k-ary trees (A000108: s=x+s^2, A001263: s=(x, y)+(x, s)+(s, y)+(s, s))), (A001764: s=x+s^3), (A002293: s=x+s^4), (A002294: s=x+s^5), (A002295: s=x+s^6), (A002296: s=x+s^7), (A007556: s=x+s^8), at most k-ary trees (A001006: s=x+xs+xs^2), (A036765-A036769, s=x+xs^2....+xs^k, k=3, 4, 5, 6, 7).

with(combinat): for n from 1 to 40 do printf(`%d,`,binomial(9*n,n)/((9-1)*n+1)) od:

G.f. A(x) satisfies: A = x + A^9.

a(n) = C(k*n, n)/((k-1)*n+1), k=9.

More terms from James Sellers, Mar 15 2001

A211248 G.f. satisfies: A(x) = (1 + xA(x)^3) (1 + x^2*A(x)^4).

Original entry on oeis.org

1, 1, 4, 20, 114, 703, 4565, 30752, 212921, 1505916, 10833164, 79018804, 583062388, 4344431508, 32641910199, 247033970128, 1881402836376, 14408753414558, 110897147057354, 857307054338476, 6653979156676983, 51831065993122915, 405060413133136902, 3175019470333290488

Offset: 0

Paul D. Hanna, Apr 05 2012

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)); here p=2 and q=1.

			G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 114*x^4 + 703*x^5 + 4565*x^6 +...
where A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 85*x^3 + 522*x^4 + 3381*x^5 + 22735*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 132*x^3 + 841*x^4 + 5588*x^5 + 38288*x^6 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 343*x^3 + 2429*x^4 + 17430*x^5 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^4 + x^3*A(x)^7.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A(x))*x*A(x)^2 + (1 + 2^2*x*A(x) + x^2*A(x)^2)*x^2*A(x)^4/2 +
(1 + 3^2*x*A(x) + 3^2*x^2*A(x)^2 + x^3*A(x)^3)*x^3*A(x)^6/3 +
(1 + 4^2*x*A(x) + 6^2*x^2*A(x)^2 + 4^2*x^3*A(x)^3 + x^4*A(x)^4)*x^4*A(x)^8/4 +
(1 + 5^2*x*A(x) + 10^2*x^2*A(x)^2 + 10^2*x^3*A(x)^3 + 5^2*x^4*A(x)^4 + x^5*A(x)^5)*x^5*A(x)^10/5 +
(1 + 6^2*x*A(x) + 15^2*x^2*A(x)^2 + 20^2*x^3*A(x)^3 + 15^2*x^4*A(x)^4 + 6^2*x^5*A(x)^5 + x^6*A(x)^6)*x^6*A(x)^12/6 +...
more explicitly,
log(A(x)) = x + 7*x^2/2 + 49*x^3/3 + 359*x^4/4 + 2706*x^5/5 + 20767*x^6/6 +...

Cf. A200716, A200717, A199876, A211249, A200731, A199874, A199877.

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

CoefficientList[Sqrt[1/x * InverseSeries[Series[x*(1 - x - x^3)^2/(1 + x^2)^2, {x, 0, 20}], x]], x] (* Vaclav Kotesovec, Nov 22 2017 *)

{a(n)=polcoeff(sqrt( (1/x)*serreverse( x*(1-x-x^3)^2/(1+x^2+x*O(x^n))^2 ) ), n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(p=2, q=1, 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=2, q=1, 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=2, q=1, 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( x*(1-x-x^3)^2/(1+x^2)^2 ) ).
(2) A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).
(3) a(n) = [x^n] ((1+x^2)/(1-x-x^3))^(2*n+2) / (n+1).
(4) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^k) * x^n*A(x)^(2*n)/n ).
(5) 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)^(2*n)/n ).
(6) A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x) = G(x*A(x)) and A(x/G(x)) = G(x) = (1 + x*G(x)^2)*(1 + x^2*G(x)^2) is the g.f. of A199874.
a(n) ~ s * sqrt((1 + 2*r*s + 3*r^2*s^4) / (3*Pi*(1 + 2*r*s + 7*r^2*s^4))) / (2*n^(3/2)*r^n), where r = 0.1194948955213353102456218138370139612914667337222... and s = 1.428770161302757679335810379290625953730830139744... are real roots of the system of equations (1 + r*s^3)*(1 + r^2*s^4) = s, r*s^2*(3 + 4*r*s + 7*r^2*s^4) = 1. - Vaclav Kotesovec, Nov 22 2017

A211249 G.f. satisfies: A(x) = (1 + xA(x)^3) (1 + x^2*A(x)^5).

Original entry on oeis.org

1, 1, 4, 21, 126, 819, 5611, 39900, 291719, 2179181, 16560175, 127617168, 994951887, 7833555324, 62196300997, 497425570173, 4003607595960, 32404662671330, 263586896132154, 2153631763231319, 17666722629907960, 145449082369322208, 1201414340736684702

Offset: 0

Paul D. Hanna, Apr 05 2012

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)); here p=2 and q=2.

			G.f.: A(x) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + 819*x^5 + 5611*x^6 +...
where A( x*(1-x-x^3)^2/(1+x^2)^2 ) = (1+x^2)/(1-x-x^3).
Related expansions:
A(x)^3 = 1 + 3*x + 15*x^2 + 88*x^3 + 564*x^4 + 3828*x^5 + 27040*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 195*x^3 + 1335*x^4 + 9486*x^5 + 69305*x^6 +...
A(x)^8 = 1 + 8*x + 60*x^2 + 448*x^3 + 3374*x^4 + 25704*x^5 +...
where A(x) = 1 + x*A(x)^3 + x^2*A(x)^5 + x^3*A(x)^8.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + x*A(x)^2)*x*A(x)^2 +
(1 + 2^2*x*A(x)^2 + x^2*A(x)^4)*x^2*A(x)^4/2 +
(1 + 3^2*x*A(x)^2 + 3^2*x^2*A(x)^4 + x^3*A(x)^6)*x^3*A(x)^6/3 +
(1 + 4^2*x*A(x)^2 + 6^2*x^2*A(x)^4 + 4^2*x^3*A(x)^6 + x^4*A(x)^8)*x^4*A(x)^8/4 +
(1 + 5^2*x*A(x)^2 + 10^2*x^2*A(x)^4 + 10^2*x^3*A(x)^6 + 5^2*x^4*A(x)^8 + x^5*A(x)^10)*x^5*A(x)^10/5 +
(1 + 6^2*x*A(x)^2 + 15^2*x^2*A(x)^4 + 20^2*x^3*A(x)^6 + 15^2*x^4*A(x)^8 + 6^2*x^5*A(x)^10 + x^6*A(x)^12)*x^6*A(x)^12/6 +...
more explicitly,
log(A(x)) = x + 7*x^2/2 + 52*x^3/3 + 403*x^4/4 + 3211*x^5/5 + 26050*x^6/6 +...

Cf. A200716, A200717, A199876, A211248, A200731, A200075, A199877.

Cf. A199874, A198951, A198953, A198957, A192415, A198888, A036765.

CoefficientList[Sqrt[1/x * InverseSeries[Series[x*(1-2*x-x^2+x^4 + (1-x-x^2) * Sqrt[(1+x+x^2)*(1-3*x+x^2)])/2, {x, 0, 20}], x]], x] (* Vaclav Kotesovec, Nov 22 2017 *)

{a(n)=polcoeff(sqrt( (1/x)*serreverse( x*(1-2*x-x^2+x^4 + (1-x-x^2)*sqrt( (1+x+x^2)*(1-3*x+x^2) +x*O(x^n)))/2 ) ), n)}
for(n=0,30,print1(a(n),", "))

{a(n)=local(p=2, q=2, 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=2, q=2, 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=2, q=2, 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.: sqrt( (1/x)*Series_Reversion( x*(1-2*x-x^2+x^4 + (1-x-x^2)*sqrt( (1+x+x^2)*(1-3*x+x^2) ))/2 ) ).
G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion(x/G(x)) where A(x) = G(x*A(x)) and A(x/G(x)) = G(x) is the g.f. of A200075.
(2) A(x) = sqrt( (1/x)*Series_Reversion( x/G(x)^2 ) ) where A(x) = G(x*A(x)^2) and G(x) = A(x/G(x)^2) and 1+x*G(x) is the g.f. of A004148.
(3) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^(2*k)) * x^n*A(x)^(2*n)/n ).
(4) A(x) = exp( Sum_{n>=1} (1-x*A(x)^2)^(2*n+1) * (Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^(2*k)) * x^n*A(x)^(2*n)/n ).
a(n) ~ s * sqrt((1 + 2*r*s^2 + 3*r^2*s^5) / (Pi*(3 + 10*r*s^2 + 28*r^2*s^5))) / (2*n^(3/2)*r^n), where r = 0.1130413665724951344267888513870607581680912144315... and s = 1.385648922830296011590145919380626723251960276539... are real roots of the system of equations (1 + r*s^3)*(1 + r^2*s^5) = s, r*s^2*(3 + 5*r*s^2 + 8*r^2*s^5) = 1. - Vaclav Kotesovec, Nov 22 2017

cp's OEIS Frontend

A369440 Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x^2)^2) ).

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A002844 Number of non-isentropic binary rooted trees with n nodes.

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A135305 Triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with k UUUU's.

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A198944 G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^k*A(x)^k] * x^n/n ).

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A303271 Number of ordered rooted trees with n non-root nodes such that the maximal outdegree equals three.

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A337512 G.f. A(x) satisfies: A(x) = 1 - Sum_{k=1..3} (x * A(x))^k.

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A378425 Expansion of (1/x) * Series_Reversion( x / (1 + x + x^2 * (1 + x)^3) ).

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A059967 Number of 9-ary trees.

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A211248 G.f. satisfies: A(x) = (1 + x*A(x)^3) * (1 + x^2*A(x)^4).

A198944 G.f.: A(x) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^3 * x^kA(x)^k] x^n/n ).

A211248 G.f. satisfies: A(x) = (1 + xA(x)^3) (1 + x^2*A(x)^4).

A211249 G.f. satisfies: A(x) = (1 + xA(x)^3) (1 + x^2*A(x)^5).