A309204 -id:A309204 - cp's OEIS frontend

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

1, 1, 13, 541, 47545, 7231801, 1695106117, 567547087381, 257320926233329, 151856004814953841, 113144789723082206461, 103890621918675777804301, 115270544419577901796226473, 152049571406030636219959644841

Offset: 0

Paul D. Hanna, Aug 26 2008, May 27 2009

nonn

			E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 541*x^6/6! + 47545*x^8/8! + ...
The e.g.f. of A007106 (a bisection of A058014) is given by:
sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 96*x^5/5! + 5888*x^7/7! + 686080*x^9/9! + ...
The e.g.f. of A058014 is given by:
F(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 96*x^5/5! + 541*x^6/6! + ...
where A(x) = [F(x) + F(-x)]/2 and exp(x*A(x)) = F(x).
The e.g.f. of A143600 is given by:
G(x) = 1 + x + 5*x^2/2! + 25*x^3/3! + 249*x^4/4! + 2561*x^5/5! + ...
where A(2x) = [G(x)/G(-x) + G(-x)/G(x)]/2.

Seiichi Manyama, Table of n, a(n) for n = 0..211

Cf. A058014, A143600, A007106, A309204.

Cf. A370430, A370432.

Table[(2*n)!*CoefficientList[1/x*InverseSeries[Series[x/Cosh[x],{x,0,41}],x],x][[2*n+1]],{n,0,20}] (* Vaclav Kotesovec, Jan 10 2014 *)

{a(n)=local(A=1+x*O(x^n));for(i=0,n,A=cosh(x*A));n!*polcoeff(A,n)}

{a(n)=(2*n)!*polcoeff(cosh(x+x*O(x^(2*n)))^(2*n+1)/(2*n+1),2*n)} \\ Paul D. Hanna, Aug 29 2008

{a(n) = sum(k=0,n, binomial(2*n+1,k) * (2*n+1-2*k)^(2*n) / ((2*n+1) * 2^(2*n)) )}
for(n=0,30, print1(a(n),", ")) \\ Paul D. Hanna, Feb 19 2024

E.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! satisfies the following formulas.
(1) A(x) = cosh(x*A(x)).
(2) A(x) = (1/x)*Series_Reversion( x/cosh(x) ).
(3) sqrt(A(x)^2 - 1) = e.g.f. of A007106.
(4) exp(x*A(x)) = A(x) + sqrt(A(x)^2-1) = e.g.f. of A058014.
(5) A(x) = [F(x) + F(-x)]/2 where F(x) = exp(x*[F(x) + 1/F(x)]/2) = e.g.f. of A058014.
(6) A(2*x) = [G(x)/G(-x) + G(-x)/G(x)]/2 where G(x) = exp(x*G(x)/G(-x)) = e.g.f. of A143600.
From Paul D. Hanna, Aug 29 2008: (Start)
(7) A(x/cosh(x)) = cosh(x).
(8) a(n) = (2n)!*[x^(2n)] cosh(x)^(2n+1)/(2n+1). (End)
(9) a(n) = Sum_{k=0..n} binomial(2*n+1,k) * (2*n+1 - 2*k)^(2*n) / ((2*n+1) * 2^(2*n)). [See formula by Christophe Vignat in A309204.] - Paul D. Hanna, Feb 19 2024
a(n) ~ 2^(2*n) * n^(2*n-1) * (s^2-1)^(n+1/2) / exp(2*n), where s = 1.810170580698977274512829... is the root of the equation sqrt(s^2-1) * log(s + sqrt(s^2-1)) = s. - Vaclav Kotesovec, Jan 10 2014
Radius of convergence r = 0.66274341934918158097474... = 1/sqrt(s^2-1) and A(r) = s (given above) satisfy r = 1/sinh(r*A(r)) and A(r) = cosh(r*A(r)). - Paul D. Hanna, Mar 04 2024

Edited by Paul D. Hanna, May 27 2009

A265011 Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx.

Original entry on oeis.org

5, 0, 6, 6, 7, 0, 9, 0, 3, 2, 1, 6, 6, 2, 2, 9, 8, 1, 9, 8, 5, 2, 5, 5, 8, 0, 4, 7, 8, 3, 5, 8, 1, 5, 1, 2, 4, 7, 2, 8, 4, 3, 5, 4, 7, 3, 4, 7, 0, 2, 0, 5, 8, 2, 9, 2, 0, 0, 0, 2, 4, 5, 8, 6, 5, 9, 4, 7, 0, 5, 1, 4, 5, 1, 3, 2, 2, 6, 9, 3, 1, 5, 0, 3

Offset: 0

John M. Campbell, Apr 06 2016

This integral has an elegant evaluation in terms of the gamma function (see below formula). There is an interesting "symmetry" between the expressions involving the gamma function in this evaluation.

			This integral is equal to 0.50667090321662298198525580478358151247...

John M. Campbell, An Algorithm for Trigonometric-Logarithmic Definite Integrals, in the Mathematica Journal, Vol. 19.10 (2017).
W. M. Gosper, Material from Bill Gosper's Computers & Math talk, M.I.T., 1989, i+38+1 pages, annotated and scanned, included with the author's permission. (There are many blank pages because about half of the original pages were two-sided, half were one-sided.) See page 8.

Decimal expansions of definite integrals over elementary functions: A256127, A256128, A256129, A204067, A204068, A205885, A206161, A206160, A206769, A229174, A083648, A094691, A098687, A177218, A188141, A233382, A256273, A258086.

Cf. A309209 (continued fraction of the negation of this constant).

Print[RealDigits[Re[Log[2] + Log[((Gamma[1 - I/2]^2 Gamma[1 + I])/(Gamma[1 + I/2]^2 Gamma[1 - I]))^(I/2)]], 10, 100]] ;
NIntegrate[Sin[Log[x]]/(x + 1)/Log[x], {x, 0, 1}]

PARI
```
intnum(x=0,1,sin(log(x))/(x+1)/log(x))
```

Equals log(2) + log(((Gamma(1 - i/2)^2*Gamma(1 + i))/(Gamma(1 + i/2)^2*Gamma(1 - i)))^(i/2)), where i = sqrt(-1) denotes the imaginary unit.

Equals Sum_{n >= 0} (-1)^n*arctan(1/(n+1)).

A309205 Denominators of coefficients of odd powers of x in expansion of f(x) = x cos (x cos (x cos( ... .

Original entry on oeis.org

1, 2, 24, 720, 8064, 3628800, 479001600, 87178291200, 2988969984000, 6402373705728000, 2432902008176640000, 1124000727777607680000, 47726800133326110720000, 21225866375084507136000000, 60977668922342772100300800000, 265252859812191058636308480000000

Offset: 1

N. J. A. Sloane, Jul 28 2019

f(x) satisfies f(x) = x * cos(f(x)), and the coefficients can be determined from this.

Alternatively, a(n) can be written in terms of (2*n)! as a(n) = (2*n)!/A309206(n).

			f(x) = x - (1/2)*x^3 + (13/24)*x^5 - (541/720)*x^7 + (9509/8064)*x^9 - (7231801/3628800)*x^11 + (1695106117/479001600)*x^13 - (567547087381/87178291200)*x^15 + ...
Coefficients are
1, -1/2, 13/24, -541/720, 9509/8064, -7231801/3628800, 1695106117/479001600, -567547087381/87178291200, 36760132319047/2988969984000, -151856004814953841/ 6402373705728000, 113144789723082206461/2432902008176640000, ...

Andrew Howroyd, Table of n, a(n) for n = 1..100
W. M. Gosper, Material from Bill Gosper's Computers & Math talk, M.I.T., 1989, i+38+1 pages, annotated and scanned, included with the author's permission. (There are many blank pages because about half of the original pages were two-sided, half were one-sided.)
W. M. Gosper, Superposition of plots -4, Jul 29 2019

Cf. A309204, A309206.

M := 20;
f := add(c[i]*z^(2*i+1),i=0..M);
f := series(f,z,2*M+3);
f2 := series(z*cos(f)-f,z,2*M+3);
for i from 0 to M do e[i]:=coeff(f2,z,2*i+1); od:
elis:=[seq(e[i],i=0..M)]; clis:=[seq(c[i],i=0..M)];
t1 := solve(elis,clis); t2 := op(t1);
t3 := subs(t2,clis);
map(denom, t3);
# Alternative:
G:= f -> f - x*cos(f):
Newt:= unapply( f - G(f)/D(G)(f),f):
ff:= x:
for k from 1 to 5 do
ff:= convert(series(Newt(ff),x,2^(k+1)),polynom)
od:
seq(denom(coeff(ff,x,2*i+1),i=0..31); # Robert Israel, Aug 18 2019

seq[n_] := Module[{p, k}, p = 1+O[x]; For[k = 2, k <= n, k++, p = Cos[x*p]]; p] // CoefficientList[#, x^2]& // Denominator;
seq[16] (* Jean-François Alcover, Sep 07 2019, from PARI *)

\\ here F(n) gives n terms of power series.
F(n)={my(p=1+O(x));for(k=2, n, p=cos(x*p)); p}
seq(n)={my(v=Vec(F(n))); vector(n, k, denominator(v[2*k-1]))} \\ Andrew Howroyd, Aug 17 2019

A309207 Continued fraction expansion of (3 tanh (3 tanh (3 tanh (...)))).

Original entry on oeis.org

2, 1, 64, 2, 1, 1, 1, 3, 4, 1, 2, 3, 1, 272213, 1, 2, 1, 16, 110, 4, 4, 4, 1, 1, 1, 4, 1, 1, 916, 21, 11, 6, 2, 2, 2, 1, 2, 16, 3, 1, 1, 2, 2, 9, 1, 2, 8, 2, 4, 3, 2, 2, 1, 9, 6, 5, 3, 1, 4, 2, 12, 17, 1, 3, 1, 3, 1, 1, 3, 1, 7, 1, 1, 2, 8, 2, 1, 1, 5, 11, 90, 1

Offset: 0

N. J. A. Sloane, Jul 29 2019

This is the continued fraction expansion of the constant defined in A309211.

W. M. Gosper, Material from Bill Gosper's Computers & Math talk, M.I.T., 1989, i+38+1 pages, annotated and scanned, included with the author's permission. (There are many blank pages because about half of the original pages were two-sided, half were one-sided.)

Cf. A309208, A309211.

ContinuedFraction[Reduce[x == 3 Tanh@x, x, Reals][[3, 2]], 33] (* Bill Gosper, Jul 30 2019 *)

More terms from Daniel Suteu, Jul 30 2019

A309208 Decimal expansion of (3 tanh (3 tanh (3 tanh (...))))^(-2).

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 8, 2, 8, 8, 7, 3, 0, 5, 8, 2, 2, 1, 9, 5, 1, 1, 2, 9, 9, 3, 1, 1, 3, 8, 8, 2, 6, 5, 4, 0, 1, 2, 0, 8, 2, 5, 4, 4, 3, 2, 3, 4, 4, 9, 7, 8, 7, 0, 1, 0, 7, 3, 5, 9, 5, 1, 4, 6, 2, 3, 4, 8, 4, 1, 0, 7, 7, 5, 3, 7, 6, 5, 2, 3, 3, 7, 9, 2, 8, 1, 3, 5, 3, 9, 7, 7, 7, 2, 6, 2, 9, 6, 5, 4, 3, 9, 1, 6, 9, 9

Offset: 0

N. J. A. Sloane, Jul 29 2019

This is the decimal expansion of 1/x^2 where x is the constant defined in A309211.

			.1122528288730582219511299311388265401208...

W. M. Gosper, Material from Bill Gosper's Computers & Math talk, M.I.T., 1989, i+38+1 pages, annotated and scanned, included with the author's permission. (There are many blank pages because about half of the original pages were two-sided, half were one-sided.)

Cf. A309207, A309211.

More terms from Daniel Suteu, Jul 30 2019

A309206 a(n) = (2*n)!/A309205(n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 1, 1, 13, 19, 5, 1, 221, 1, 1, 1, 1, 1, 13, 17, 5, 1, 47, 4913, 29, 7, 11, 53, 1, 47, 325, 13, 1147, 41, 1, 1, 41, 1081, 11, 1, 5, 1, 1, 83, 1, 1, 133, 1, 2491, 97, 5, 103, 61, 1, 1, 19, 226493, 1, 1, 1, 5, 31, 1, 1, 1, 1271, 289

Offset: 0

N. J. A. Sloane, Jul 28 2019, following a suggestion from Bill Gosper

Bill Gosper points out that this is a better fingerprint for the series than A309205.

Cf. A309204, A309205.

F[n_] := Module[{p}, p = 1 + O[x]; For[k=2, k <= n, k++, p = Cos[x p]]; p];
seq[n_] := Module[{v}, v = CoefficientList[F[n], x]; Table[(2(k - 1))!/ Denominator[v[[2k - 1]]], {k, 1, n}]];
seq[71] (* Jean-François Alcover, Aug 27 2019, from PARI *)

\\ here F(n) gives n terms of power series.
F(n)={my(p=1+O(x)); for(k=2, n, p=cos(x*p)); p}
seq(n)={my(v=Vec(F(n))); vector(n, k, (2*(k-1))!/denominator(v[2*k-1]))} \\ Andrew Howroyd, Aug 17 2019

Terms a(31) and beyond from Andrew Howroyd, Aug 17 2019

A309209 Continued fraction expansion of the negation of the constant defined by A265011 (among other formulas, this is Sum_{k >= 1} arctan((-1)^k/k)).

Original entry on oeis.org

-1, 2, 36, 1, 40, 1, 94, 1, 1

Offset: 0

N. J. A. Sloane, Jul 29 2019; definition corrected Jul 30 2019 (Thanks to Jianing Song for pointing out that something was wrong.)

W. M. Gosper, Material from Bill Gosper's Computers & Math talk, M.I.T., 1989, i+38+1 pages, annotated and scanned, included with the author's permission. (There are many blank pages because about half of the original pages were two-sided, half were one-sided.) See page 8.

Cf. A265011.

Name corrected by Jianing Song, Aug 30 2019

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A265011 Decimal expansion of Integral_{x=0..1} sin(log(x))/((x+1)*log(x)) dx.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A309205 Denominators of coefficients of odd powers of x in expansion of f(x) = x cos (x cos (x cos( ... .

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A309207 Continued fraction expansion of (3 tanh (3 tanh (3 tanh (...)))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A309208 Decimal expansion of (3 tanh (3 tanh (3 tanh (...))))^(-2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A309206 a(n) = (2*n)!/A309205(n).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A309209 Continued fraction expansion of the negation of the constant defined by A265011 (among other formulas, this is Sum_{k >= 1} arctan((-1)^k/k)).

Views

Author

Keywords

Links

Crossrefs

Extensions