A138764 -id:A138764 - cp's OEIS frontend

A058014 Number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

1, 1, 1, 4, 13, 96, 541, 5888, 47545, 686080, 7231801, 130179072, 1695106117, 36590059520, 567547087381, 14290429935616, 257320926233329, 7405376630685696, 151856004814953841, 4917457306800619520, 113144789723082206461, 4071967909087792857088

Offset: 0

Alex Postnikov (apost(AT)math.mit.edu), Nov 13 2000

			E.g.f.: A(x) = 1 + x + x^2/2! + 4x^3/3! + 13x^4/4! + 96x^5/5! +...

Seiichi Manyama, Table of n, a(n) for n = 0..422
Alexander Postnikov, Papers.
A. Postnikov and R. P. Stanley, Deformations of Coxeter hyperplane arrangements, J. Combin. Theory, Ser. A, 91 (2000), 544-597. (Section 10.2.)

Cf. bisections: A007106, A143601.

Cf. A138764 (variant).

a := n -> 2^(-n)*add(binomial(n,k)*(n+1-2*k)^(n-1), k=0..n);

a[n_] := Sum[((n-2k+1)^(n-1)*n!) / (k!*(n-k)!), {k, 0, n}] / 2^n; a[1] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 14 2011, after Maple *)

{a(n)=local(A=1+x);for(i=0,n,A=exp(x*(A+1/(A +x*O(x^n)))/2));n!*polcoeff(A,n)} \\ Paul D. Hanna, Mar 29 2008

{a(n) = sum(k=0, n, binomial(n, k)*(n+1-2*k)^(n-1))/2^n} \\ Seiichi Manyama, Sep 27 2020

a(n) = (1/2^n) * Sum_{k=0..n} binomial(n,k) * (n + 1 - 2*k)^(n-1).
From Paul D. Hanna, Mar 29 2008: (Start)
E.g.f. satisfies A(x) = exp( x*[A(x) + 1/A(x)]/2 ).
E.g.f. A(x) equals the inverse function of 2*x*log(x)/(1 + x^2).
Let r = radius of convergence of A(x), then r = 0.6627434193491815809747420971092529070562335491150224... and A(r) = 3.31905014223729720342271370055697247448941708369151595... where A(r) and r satisfy A(r) = exp( (A(r)^2 + 1)/(A(r)^2 - 1) ) and r = 2*A(r)/(A(r)^2 - 1). (End)
E.g.f. A(x)=exp(B(x)), B(x) satisfies B(x)=x*cosh(B(x)). [Vladimir Kruchinin, Apr 19 2011]
a(n) ~ (1-(-1)^n*s^2)/s * n^(n-1) * ((1-s^2)/(2*s))^n / exp(n), where s = 0.3012910191606573456... is the root of the equation (1+s^2) = (s^2-1)*log(s), r = 2*s/(1-s^2). - Vaclav Kotesovec, Jan 08 2014
E.g.f. satisfies A(-x) = 1/A(x). - Alexander Burstein, Oct 26 2021
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (2*k+1) * binomial(n-1,2*k) * a(2*k) * a(n-1-2*k). - Seiichi Manyama, Jul 05 2025

Updated URL and author's e-mail address - R. J. Mathar, May 23 2010

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

A372232 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * (1 + A(x)^(1/2)) ).

Original entry on oeis.org

1, 4, 40, 688, 17152, 564864, 23212288, 1145627648, 66082594816, 4365282304000, 325074868781056, 26950224851927040, 2462208223872286720, 245811899064585814016, 26626175172644096180224, 3110339882223194198769664, 389786352057654976473726976

Offset: 0

Seiichi Manyama, Apr 23 2024

nonn

Cf. A138764, A372177.

a(n, r=2, t=1, u=1) = r*sum(k=0, n, (t*n+u*k+r)^(n-1)*binomial(n, k));

E.g.f.: A(x) = B(x)^2 where B(x) is the e.g.f. of A138764.
If e.g.f. satisfies A(x) = exp( r*x*A(x)^(t/r) * (1 + A(x)^(u/r)) ), then a(n) = r * Sum_{k=0..n} (t*n+u*k+r)^(n-1) * binomial(n,k).
a(n) = 2^n * A372177(n).

A372250 E.g.f. A(x) satisfies A(x) = exp( x * A(x) * (1 + x^2 * A(x)) ).

Original entry on oeis.org

1, 1, 3, 22, 221, 2796, 44527, 857074, 19250457, 494672824, 14331709691, 462277166814, 16430392693717, 638087135544340, 26885810437578471, 1221643263680551786, 59548603124180235953, 3099741728641090867056, 171619335190080336331891

Offset: 0

Seiichi Manyama, Apr 24 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..376

Cf. A088695, A138764.

Cf. A362392.

a(n) = n!*sum(k=0, n\3, (n-k+1)^(n-2*k-1)/(k!*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/3)} (n-k+1)^(n-2*k-1) / (k! * (n-3*k)!).

cp's OEIS Frontend

A058014 Number of labeled trees with n+1 nodes such that the degrees of all nodes, excluding the first node, are odd.

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

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A372232 E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) * (1 + A(x)^(1/2)) ).

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A372250 E.g.f. A(x) satisfies A(x) = exp( x * A(x) * (1 + x^2 * A(x)) ).

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