A036778 -id:A036778 - 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

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

A216187 Number of labeled rooted trees on n nodes such that each internal node has an odd number of children.

Original entry on oeis.org

0, 1, 2, 6, 28, 200, 1926, 22512, 306104, 4770432, 84234250, 1663735040, 36320155092, 867963393024, 22535294920334, 631718010255360, 19016907901995376, 611869203759792128, 20954324710009221138, 761015341362413371392, 29214930870257449355660

Offset: 0

Geoffrey Critzer, Mar 11 2013

nonn

			a(5) = 200: There are three unlabeled rooted trees of 5 nodes with all internal nodes having an odd number of children. They can be labeled respectively in 20 + 120 + 60 = 200 ways.
  ..o............o............o....
  ..|............|.........../|\...
  ..o............o..........o.o.o..
  ./|\...........|..........|......
  o.o.o..........o..........o......
  ...............|.................
  ...............o.................
  ...............|.................
  ...............o.................

Alois P. Heinz, Table of n, a(n) for n = 0..150

Cf. A036778.

a:= n-> n!*coeff(series(RootOf(F=x*(sinh(F)+1), F), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 12 2013

nn=12; f[x_]:=Sum[a[n]x^n/n!, {n,0,nn}]; s=SolveAlways[0==Series[f[x]-x (Sinh[f[x]]+1), {x,0,nn}], x]; Table[a[n], {n,0,nn}]/.s

E.g.f. satisfies: F(x) = x*(sinh(F(x))+1).

a(n) ~ sqrt(s/(s-r)) * n^(n-1) / (exp(n) * r^n), where r = 0.482309923717218507261475229723265094762759829863... and s = 1.358310572965774067065006624540704170183889018218... are real roots of the system of equations s = r*(1 + sinh(s)), r*cosh(s) = 1. - Vaclav Kotesovec, Jun 07 2021

A263547 E.g.f. satisfies: A(x) = exp( x * real( A(x)^I ) ), where I^2 = -1.

Original entry on oeis.org

1, 1, 1, -2, -11, 36, 421, -1896, -35223, 201232, 5188201, -35856160, -1188970595, 9633456704, 391498316301, -3636762088064, -175238714193967, 1835360835895552, 102369229796454481, -1193179646751072768, -75645902492063337659, 971018266973866894336, 68985480327663686993141, -966900537026209266460672

Offset: 0

Paul D. Hanna, Oct 20 2015

sign

			E.g.f.: A(x) = 1 + x + x^2/2! - 2*x^3/3! - 11*x^4/4! + 36*x^5/5! + 421*x^6/6! - 1896*x^7/7! - 35223*x^8/8! + 201232*x^9/9! + 5188201*x^10/10! +...
where
log(A(x)) = x - 3*x^3/3! + 65*x^5/5! - 3787*x^7/7! + 427905*x^9/9! - 79549811*x^11/11! +...+ A036778(n)*x^(2*n-1)/(2*n-1)! +...
which equals Series_Reversion( x/cos(x) ).
Also,
A(x)^I = 1 + I*x - x^2 - 4*I*x^3 + 13*x^4 + 96*I*x^5 - 541*x^6 - 5888*I*x^7/7! + 47545*x^8/8! +...+ A058014(n)*I^n*x^n/n! +...
Further,
Series_Reversion(A(x)-1) = log(1+x)/cos(log(1+x)) = e.g.f. of A009424.

Cf. A036778, A058014.

{a(n) = my(A=1); for(i=1,n+1, A = exp(x*real(A^I) +x*O(x^n))); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{a(n) = my(A=1); for(i=1,n+1, A = exp( serreverse( x/cos(x +x*O(x^n))))); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: A(x) = exp( Series_Reversion( x/cos(x) ) ).

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A216187 Number of labeled rooted trees on n nodes such that each internal node has an odd number of children.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A263547 E.g.f. satisfies: A(x) = exp( x * real( A(x)^I ) ), where I^2 = -1.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

PARI

Formula