A058014 -id:A058014 - cp's OEIS frontend

A007106 Number of labeled odd degree trees with 2n nodes.

1, 4, 96, 5888, 686080, 130179072, 36590059520, 14290429935616, 7405376630685696, 4917457306800619520, 4071967909087792857088, 4113850542422629363482624, 4980673081258443273955966976, 7119048451600750435732824260608, 11861520124846917915630931846103040

Offset: 1

N. J. A. Sloane

nonn

			From _Peter Bala_, Apr 24 2012: (Start)
Let G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + ... be the e.g.f. for A143601. Then sinh(x*G(x)) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
Conjectural e.g.f. as an x-adic limit:
sinh(x) = x + ...; sinh(x*cosh(x)) = x + 4*x^3/3! + ...;
sinh(x*cosh(x*cosh(x))) = x + 4*x^3/3! + 96*x^5/5! + ...;
sinh(x*cosh(x*cosh(x*cosh(x)))) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + ....
(End)

R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..211 (terms 1..39 from R. W. Robinson)
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
Mathematics Stack Exchange, Marko R. Riedel, Odd degree trees
Mathematics Stack Exchange, Marko R. Riedel, Odd degree trees II
Marko Riedel, Count by Prüfer codes and Stirling numbers
Index entries for sequences related to trees

Cf. A036778, A058014, A060279, A143601, A156289.

A007106(n) = A(2n) where n>=2, A(n) = (add(binomial(n,q)*(n-2*q)^(n-2)/(n-2)!, q=0..n) - add(binomial(n-1,q)*(n-2*q)^(n-3)/(n-3)!, q=0..n-1) + add(binomial(n-1,q)*(n-2-2*q)^(n-3)/(n-3)!, q=0..n-1))*n!/2^(n+1)/(n-1)

{1}~Join~Array[(1/2)*Sum[Binomial[2 #, k]*(# - k)^(2 # - 2), {k, 0, # - 1}] &, 12, 2] (* Michael De Vlieger, Oct 13 2021 *)

a(n) = if(n<=1, n==1, sum(k=0, n-1, binomial(2*n,k) * (n-k)^(2*n-2))/2) \\ Andrew Howroyd, Nov 22 2021

a(n) = A060279(n)/(2*n). - Vladeta Jovovic, Feb 08 2005
Bisection of A058014. The expansion 1/sqrt(1+x^2)*arcsinh(x) = x - 4*x^3/3! + 64*x^5/5! - ... (see A002454) has series reversion x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + .... The coefficients appear to be the terms of this sequence. As an x-adic limit this e.g.f. equals lim_{n -> infinity} sinh(f(n,x)), where f(0,x) = x and f(n,x) = x*cosh(f(n-1,x)) for n >= 1. See the example section below. - Peter Bala, Apr 24 2012
a(n) = Sum_{k=1..n} binomial(n,k) * k! * (n-2)! [z^{n-2}] [u^k] exp(u(exp(z)+exp(-z)-2)/2)). - Marko Riedel, Jun 16 2016
From Alexander Burstein, Oct 13 2021: (Start)
a(n) = (1/2) * Sum_{k=0..n-1} binomial(2*n,k) * (n-k)^(2*n-2) for n >= 2.
a(n) = (2*n-1)!*[x^(2*n-1)] sinh(REVERT(x/cosh(x))), see A036778. (End)
a(n) = Sum_{k=0..n-1} A156289(n-1, k)*(2*n)!/(2*n-k)!. - Peter Luschny, May 07 2022

Corrected and extended by Vladeta Jovovic, Feb 08 2005

A202617 E.g.f. satisfies: A(x) = exp( x*(1 + A(x)^2)/2 ).

Original entry on oeis.org

1, 1, 3, 19, 185, 2441, 40747, 823691, 19564785, 534145105, 16482667091, 567343245635, 21552042260905, 895664877901145, 40422799315249275, 1968883362773653051, 102942561775293158369, 5750760587905912310177, 341848844954020959953059, 21545207157567497255044979

Offset: 0

Paul D. Hanna, Dec 21 2011

nonn

Compare to e.g.f. F(x) of A007889: F(x) = exp(x*(1 + F(x))/2), where A007889(n) = number of intransitive (or alternating) trees: vertices are [0,n] and for no i

Related sequence: A058014(n) = number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 185*x^4/4! + 2441*x^5/5! +...
where log(A(x)) = x*(1 + A(x)^2)/2 and
log(A(x)) = x + 2*x^2/2! + 12*x^3/3! + 112*x^4/4! + 1440*x^5/5! + 23616*x^6/6! +...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 56*x^3/3! + 576*x^4/4! + 7872*x^5/5! + 134656*x^6/6! +...

Seiichi Manyama, Table of n, a(n) for n = 0..372

Cf. A007889, A058014, A214225, A138860, A202357.

CoefficientList[Series[Sqrt[-ProductLog[-E^x*x]/x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 10 2014 *)

a(n)=local(A=1+x); for(i=0, n, A=exp(x*(1+A^2)/2 +x*O(x^n))); n!*polcoeff(A, n)

/* Coefficients of A(x)^p are given by: */
{a(n, p=1)=(1/2^n)*sum(k=0, n, binomial(n, k)*p*(2*k+p)^(n-1))}

a(n)=n!*polcoeff(exp(sum(k=1,n,k^(k-1)*cosh(k*x +x*O(x^n))*x^k/k!) +x*O(x^n)),n)
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 20 2012

E.g.f. A(x) equals the formal inverse of function 2*log(x)/(1+x^2).
E.g.f.: exp( Sum_{n>=1} n^(n-1) * cosh(n*x) * x^n / n! ). - Paul D. Hanna, Nov 20 2012
E.g.f.: exp(G(x)) where G(x) = x/(1 - tanh(G(x))) is the e.g.f. of A214225. - Paul D. Hanna, Nov 20 2012
E.g.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the e.g.f. of A058014.
a(n) = (1/2^n)*Sum_{k=0..n} C(n,k)*(2*k+1)^(n-1).
Powers of e.g.f.:
If A(x)^p = Sum_{n>=0} a(n,p)*x^n/n! then a(n,p) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*p*(2*k+p)^(n-1).
a(n) ~ sqrt(1+c) * n^(n-1) / (2 * exp(n) * c^(n+1/2)), where c = LambertW(exp(-1)) = 0.278464542761... (see A202357). - Vaclav Kotesovec, Jan 10 2014
E.g.f.: sqrt(-LambertW(-x*exp(x))/x). - Vaclav Kotesovec, Jan 10 2014

A138860 E.g.f. satisfies: A(x) = exp( x*(A(x) + A(x)^2)/2 ).

Original entry on oeis.org

1, 1, 4, 31, 364, 5766, 115300, 2788724, 79197040, 2583928360, 95256535936, 3916137470664, 177651980724160, 8815348234689920, 474993826614917632, 27619367979975064576, 1723821221240101984000, 114948301218300412117632

Offset: 0

Paul D. Hanna, Apr 01 2008, Apr 02 2008, Apr 03 2008

nonn

The related sequence A007889 enumerates the number of intransitive (or alternating) trees.

a(n+1) is the number of incomplete ternary trees on n labeled vertices in which each left child has a larger label than its parent and each middle child has a smaller label than its parent. - Brian Drake, Jul 28 2008

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 364*x^4/4! + 5766*x^5/5! + ...

Seiichi Manyama, Table of n, a(n) for n = 0..363

Cf. A007889, A088789, A058014, A036778, A138903.

Cf. A138764.

Table[1/2^n * Sum[Binomial[n,k]*(n+k+1)^(n-1),{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 15 2013 *)

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

/* Series Reversion: */
a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp(serreverse(2*x/(exp(X)+exp(2*X)) )),n)

/* Coefficients of A(x)^p are given by: */
{a(n,p=1)=(1/2^n)*sum(k=0,n,binomial(n,k)*p*(n+k+p)^(n-1))}

a(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+k+1)^(n-1) - Vladeta Jovovic, Mar 31 2008.
E.g.f. satisfies: A( 2*x/( exp(x) + exp(2*x) ) ) = exp(x).
E.g.f.: A(x) = inverse function of 2*log(x)/(x + x^2).
E.g.f.: A(x) = exp( Series_Reversion[ 2*x/(exp(x) + exp(2*x)) ] ).
E.g.f.: A(x) = G(x/2) where G(x) = e.g.f. of A138764.
More generally, if A(x) = Sum_{n>=0} a(n)*x^n/n! = exp( x*[A(x) + A(x)^m]/2 ) then a(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+(m-1)*k+1)^(n-1) and if B(x) = Sum_{n>=0} b(n)*x^n/n! = log(A(x)) then b(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+(m-1)*k)^(n-1). - Paul D. Hanna and Vladeta Jovovic, Apr 02 2008
Powers of e.g.f.: If A(x)^p = Sum_{n>=0} a(n,p)*x^n/n! then
. a(n,p) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*p*(n+k+p)^(n-1).
Given e.g.f. A(x), let B(x) = e.g.f. of A007889, then
. A(x) = B(x*A(x)) = (1/x)*Series_Reversion(x/B(x)) and
. B(x) = A(x/B(x)) = x/Series_Reversion(x*A(x)).
a(n) ~ n^(n-1)*(1+r)^n*r^(n+1)/(sqrt(1+3*r)*(1-r)^(2*n+1)*exp(n)*2^n), where r = 0.6472709258412625... is the root of the equation (r/(1-r))^(1+r) = e. - Vaclav Kotesovec, Jun 15 2013

A143600 E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)).

Original entry on oeis.org

1, 1, 5, 25, 249, 2561, 40573, 641817, 14110001, 302279617, 8530496181, 230851019609, 7964867290537, 260618470319169, 10635790073585069, 408342804482252761, 19246730825243728737, 848289638051491455617, 45356940470607637151845, 2257054105205570995111833

Offset: 0

Paul D. Hanna, Aug 26 2008

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! +...
A LambertW identity yields the series:
A(x) = 1 + x/A(-x) + 3^1*x^2/2!/A(-x)^2 + 4^2*x^3/3!/A(-x)^3 + 5^3*x^4/4!/A(-x)^4 + 6^4*x^5/5!/A(-x)^5 +...+ (n+1)^(n-1)*x^n/n!/A(-x)^n +...
RELATED EXPANSIONS.
A(x)/A(-x) = F(2x) where F(x) is the e.g.f. of A058014:
A(x)/A(-x) = 1 + 2*x + 4*x^2/2! + 32*x^3/3! + 208*x^4/4! + 3072*x^5/5! +...
F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
which satisfies: F(x) = exp(x*(F(x) + 1/F(x))/2).
(A(x)/A(-x) + A(-x)/A(x))/2 = G(2x) where G(x) is the e.g.f. of A143601:
(A(x)/A(-x) + A(-x)/A(x))/2 = 1 + 4*x^2/2! + 208*x^4/4! + 34624*x^6/6! +...
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
which satisfies G(x) = cosh(x*G(x)).

Vaclav Kotesovec, Table of n, a(n) for n = 0..175

Cf. A058014, A143601, A007106.

a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x*A/subst(A,x,-x)));n!*polcoeff(A,n)

/* Formula Using a LambertW Identity: */
{a(n)=local(A=1);for(i=1,n,A=sum(k=0,n,(k+1)^(k-1)*x^k/k!/subst(A,x,-x)^k+x*O(x^n)));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Nov 05 2012

E.g.f. A(x) satisfies:
(1) A(x) = exp(x*exp(2x*G(2x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
(2) [A(x)/A(-x) + A(-x)/A(x)]/2 = G(2x) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
(3) A(x)/A(-x) = exp(x*[A(x)/A(-x) + A(-x)/A(x)]) = F(2x) where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
(4) A(x) = Sum_{n>=0} (n+1)^(n-1) * x^n/n! / A(-x)^n.
(5) A(x)^m = Sum_{n>=0} m*(n+m)^(n-1) * x^n/n! / A(-x)^n.
(6) log(A(x)) = Sum_{n>=1} n^(n-1) * x^n/n! / A(-x)^n = x*A(x)/A(-x).
Formulas (4), (5), and (6) are due to LambertW identities. - Paul D. Hanna, Nov 05 2012
a(n) ~ c * n! / (n^(3/2) * r^n), where r = 0.33137170967459079... is the root of the equation sqrt(1+4*r^2) = log((1+sqrt(1+4*r^2))/(2*r)), and c = 1.35397895306096963692514418... if n is even, and c = 1.281887793570420328585518150... if n is odd. - Vaclav Kotesovec, Feb 25 2014

A143599 E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ).

Original entry on oeis.org

1, 1, 3, 10, 53, 316, 2527, 22072, 239689, 2774800, 38284091, 553477024, 9284250109, 161180444608, 3187413648343, 64638167906176, 1473221217774353, 34190645940363520, 882759869810501491, 23079229227696318976

Offset: 0

Paul D. Hanna, Aug 27 2008

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 316*x^5/5! +...
F(x) = sqrt(A(x)/A(-x)) = e.g.f. of A058014:
F(x) = 1 + x + 1*x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! +...
where F(x) = exp(x*(F(x) + 1/F(x))/2).
G(x) = [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601:
G(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! +...
where G(x) = cosh(x*G(x)).
S(x) = [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106:
S(x) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! +...
where S(x) = sqrt(G(x)^2 - 1) and G(x) = e.g.f. of A143601.

Vaclav Kotesovec, Table of n, a(n) for n = 0..218

Cf. A058014, A143600, A143601, A007106.

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=exp(x*sqrt(A/subst(A,x,-x))));n!*polcoeff(A,n)}

E.g.f.: A(x) = exp(x*exp(x*G(x))) where G(x) = cosh(x*G(x)) = e.g.f. of A143601.
E.g.f.: sqrt(A(x)/A(-x)) = F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
E.g.f.: [sqrt(A(x)/A(-x)) + sqrt(A(-x)/A(x))]/2 = e.g.f. of A143601.
E.g.f.: [sqrt(A(x)/A(-x)) - sqrt(A(-x)/A(x))]/2 = e.g.f. of A007106.
E.g.f.: A(x) = H(x/2)^2 where H(x) = exp(x*H(x)/H(-x)) = e.g.f. of A143600.
E.g.f. satisfies: A(x/cosh(x)) = exp(x*exp(x)/cosh(x)). [From Paul D. Hanna, Aug 29 2008]
E.g.f. satisfies: -x*sqrt(x^2/log(y)^2) = log((x^2*y)/log(y)^2), where y=A(x). - Vaclav Kotesovec, Feb 26 2014
a(n) ~ c * n! * d^n / n^(3/2), where d = 1.5088795615383199289... is the root of the equation sqrt(1+1/d^2) = 1 + LambertW((1+sqrt(1+1/d^2))/exp(1 + sqrt(1+1/d^2))), and c = 7.98255033020099890281693169... if n is even, and c = 7.852067808737280621088934789... if n is odd. - Vaclav Kotesovec, Feb 26 2014

A199202 E.g.f. satisfies: A(x) = exp( x*(A(x) + 1/A(-x))/2 ).

Original entry on oeis.org

1, 1, 3, 10, 53, 376, 3607, 38032, 498409, 7122304, 121691051, 2182921984, 45592175389, 987527547904, 24479592884671, 620921169012736, 17795726532904913, 517636848366223360, 16851227968120051027, 552890360903850459136, 20150074601540899828741

Offset: 0

Paul D. Hanna, Nov 03 2011

nonn

Compare to the e.g.f. G(x) of A058014, which satisfies both: G(x) = exp(x*(G(x) + 1/G(x))/2) and G(x) = exp(x*(G(x) + G(-x))/2); A058014 counts labeled trees such that the degrees of all nodes, excluding the first, are odd.

			E.g.f.:  A(x) = 1 + x + 3*x^2/2! + 10*x^3/3! + 53*x^4/4! + 376*x^5/5! +.. .
Let B(x) = log(A(x))/x = (A(x) + 1/A(-x))/2 then B(x) begins:
B(x) = 1 + x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 216*x^5/5! + 1561*x^6/6! + 19328*x^7/7! +...+ A198198(n)*x^n/n! +...
such that B(x) = (exp(x*B(x)) + exp(x*B(-x)))/2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..320

Cf. A198198, A058014.

{a(n)=local(A=1+x*O(x^n)); for(n=0, n, A=exp(x*(A+1/subst(A, x, -x))/2+x*O(x^n))); n!*polcoeff(A, n)}

E.g.f.: A(x) = exp(x*B(x)) where B(x) = (exp(x*B(x)) + exp(x*B(-x)))/2 is the e.g.f. of A198198.
E.g.f. satisfies: log(x) = x*log(y)/(x*y^2 - 2*y*log(y)) + log(2*log(y) - x*y), where y = A(x). - Vaclav Kotesovec, Feb 28 2014
a(n) ~ c * n! * d^n / n^(3/2), where d = 1.9126860724609002014... (see A198198), and c = 1.84843299011729... if n is even, and c = 1.808309580980992... if n is odd. - Vaclav Kotesovec, Feb 28 2014

A385687 E.g.f. A(x) satisfies A(x) = exp( x*((A(x) + A(-x))/2)^2 ).

Original entry on oeis.org

1, 1, 1, 7, 25, 341, 2161, 44115, 404209, 11010025, 132273601, 4508793983, 67085545033, 2747071330173, 48765277295281, 2331905267846731, 48106649137922017, 2631174441142423505, 61862217319644572161, 3809106344377237185399, 100542158725584301036921

Offset: 0

Seiichi Manyama, Jul 06 2025

nonn

Cf. A058014, A385688.

Cf. A143546, A360987, A385690.

terms = 21;  A[] = 1; Do[A[x] = Exp[x*((A[x] + A[-x])/2)^2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 07 2025 *)

E.g.f. A(x) satisfies A(-x) = 1/A(x).

a(0) = 1; a(n) = (n-1)! * Sum_{i, j, k>=0 and i+2*j+2*k=n-1} (n-i) * a(i) * a(2*j) * a(2*k)/(i! * (2*j)! * (2*k)!).

A385691 E.g.f. A(x) satisfies A(x) = exp( x(A(x) + A(wx) + A(w^2x))/3 ), where w = exp(2Pi*i/3).

Original entry on oeis.org

1, 1, 1, 1, 5, 21, 61, 568, 4257, 20917, 286451, 3099141, 21555865, 390273898, 5524889553, 49790422501, 1121734897937, 19631020478229, 217441607213557, 5862333450708460, 122222268766006641, 1606671304363320805, 50443794604147639487, 1220712011020970521461

Offset: 0

Seiichi Manyama, Jul 07 2025

nonn

Cf. A058014, A124753.

terms = 24;  w = Exp[2*Pi*I/3]; A[] = 1; Do[A[x] = Exp[x*(A[x] + A[w*x] + A[w^2*x])/3] + O[x]^terms // Normal, terms]; Simplify[CoefficientList[A[x], x]Range[0,terms-1]!] (* Stefano Spezia, Jul 07 2025 *)

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} (3*k+1) * binomial(n-1,3*k) * a(3*k) * a(n-1-3*k).

A385620 E.g.f. A(x) satisfies A(x) = exp( x(A(2x) + A(3*x)) ).

Original entry on oeis.org

1, 2, 24, 1064, 158144, 78427712, 130391102464, 725657074158592, 13450842239318679552, 825492067428121929359360, 166724642619378284453845213184, 110175812687250637947409895640473600, 236918101449618886434191300434062010777600, 1649425480856495624442166311045759714226010423296

Offset: 0

Seiichi Manyama, Jul 05 2025

nonn

Cf. A058014, A096538, A385526.

terms = 14; A[] = 1; Do[A[x] =Exp[x*(A[2*x] + A[3*x])]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 05 2025 *)

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (j+1)*(2^j+3^j)*binomial(i-1, j)*v[j+1]*v[i-j])); v;

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (2^k+3^k) * binomial(n-1,k) * a(k) * a(n-1-k).

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cp's OEIS Frontend

A007106 Number of labeled odd degree trees with 2n nodes.

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A143601 Number of labeled odd-degree trees with 2n+1 nodes.

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A202617 E.g.f. satisfies: A(x) = exp( x*(1 + A(x)^2)/2 ).

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A138860 E.g.f. satisfies: A(x) = exp( x*(A(x) + A(x)^2)/2 ).

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A143600 E.g.f. satisfies: A(x) = exp(x*A(x)/A(-x)).

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A143599 E.g.f. satisfies: A(x) = exp( x*sqrt(A(x)/A(-x)) ).

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A199202 E.g.f. satisfies: A(x) = exp( x*(A(x) + 1/A(-x))/2 ).

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A385691 E.g.f. A(x) satisfies A(x) = exp( x(A(x) + A(wx) + A(w^2x))/3 ), where w = exp(2Pi*i/3).

A385620 E.g.f. A(x) satisfies A(x) = exp( x(A(2x) + A(3*x)) ).