A124214 -id:A124214 - cp's OEIS frontend

A124212 Expansion of e.g.f. exp(x)/sqrt(2-exp(2*x)).

1, 2, 8, 56, 560, 7232, 114368, 2139776, 46223360, 1132124672, 30999600128, 938366468096, 31114518056960, 1121542540992512, 43664751042265088, 1826043989622358016, 81635676596544143360

Offset: 0

Karol A. Penson, Oct 19 2006

Robert Israel, Table of n, a(n) for n = 0..60

Cf. A000629, A000984, A014307, A124214, A145901, A176785, A186695, A229558.

Cf. A276371, A365777.

  N:= 60; # to get a(n) for n <= N
S:= series(exp(x)/sqrt(2-exp(2*x)), x, N+1):
seq(coeff(S,x,j), j=0..N); # Robert Israel, May 19 2014

CoefficientList[Series[E^x/Sqrt[2-E^(2*x)]-1, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 03 2013 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=0,n,A=1+intformal(A+A^3)); n!*polcoeff(A,n)} \\ Paul D. Hanna, Oct 04 2008

E.g.f. satisfies: A'(x) = A(x) + A(x)^3 with A(0)=1. [From Paul D. Hanna, Oct 04 2008]
G.f.: 1/G(0) where G(k) = 1 - x*(4*k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 4*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(n) ~ 2^(n+1/2)*n^n/(log(2)^(n+1/2)*exp(n)). - Vaclav Kotesovec, Jun 03 2013
From Peter Bala, Aug 30 2016: (Start)
a(n) = 1/sqrt(2) * Sum_{k >= 0} (1/8)^k*binomial(2*k,k)*(2*k + 1)^n = 1/sqrt(2) * Sum_{k >= 0} (-1/2)^k*binomial(-1/2,k)*(2*k + 1)^n. Cf. A176785, A124214 and A229558.
a(n) = Sum_{k = 0..n} (1/4)^k*binomial(2*k,k)*A145901(n,k).
a(n) = Sum_{k = 0..n} ( Sum_{i = 0..k} (-1)^(k-i)/4^k* binomial(2*k,k)*binomial(k,i)*(2*i + 1)^n ). (End)
a(n) = 2^n * A014307(n). - Seiichi Manyama, Nov 18 2023

Definition corrected by Robert Israel, May 19 2014

A229558 E.g.f.: exp(x) / (2 - exp(4*x))^(1/4).

Original entry on oeis.org

1, 2, 12, 152, 2832, 69152, 2089152, 75204992, 3142025472, 149428961792, 7969790856192, 471098477484032, 30567292903821312, 2159857294035525632, 165083372031671058432, 13570774387950150582272, 1193933787763434969956352, 111932230270819401046556672

Offset: 0

Paul D. Hanna, Dec 18 2013

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 152*x^3/3! + 2832*x^4/4! + 69152*x^5/5! +...
where A(x)^5 = 1 + 10*x + 140*x^2/2! + 2680*x^3/3! + 66320*x^4/4! +...
Also, A(x)^4 = 1 + 8*x + 96*x^2/2! + 1664*x^3/3! + 38400*x^4/4! +...
and log(A(x)) = 2*x + 8*x^2/2! + 96*x^3/3! + 1664*x^4/4! + 38400*x^5/5! +...

G. C. Greubel, Table of n, a(n) for n = 0..345

Cf. A124214, A034385, A124212.

CoefficientList[Series[E^x/(2-E^(4*x))^(1/4), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 19 2013 *)

{a(n)=local(A=1+x,X=x+x*O(x^n));n!*polcoeff(exp(X)/(2-exp(4*X))^(1/4),n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A+A^5+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies: A'(x) = A(x) + A(x)^5.
E.g.f. A(x) satisfies: A(x) = exp(x + Integral A(x)^4 dx).
a(n) ~ GAMMA(3/4) * 4^n * n^(n-1/4) / (sqrt(Pi) * exp(n) * log(2)^(n+1/4)). - Vaclav Kotesovec, Dec 19 2013
a(n) = 1/2^(1/4) * Sum_{k >= 0} (1/32)^k*A034385(k)*(4*k + 1)^n = 1/2^(1/4)*Sum_{k >= 0} (-1/2)^k*binomial(-1/4, k)*(4*k + 1)^n. Cf. A124212 and A124214. - Peter Bala, Aug 30 2016

A228608 E.g.f. A(x) satisfies: A'(x) = A(x)^2 + A(x)^4.

Original entry on oeis.org

1, 2, 12, 128, 1968, 39488, 977088, 28742912, 979744512, 37968868352, 1648597834752, 79272057049088, 4181485522464768, 240067201819885568, 14902137637759008768, 994529776192394166272, 71009035425186633940992, 5401058272888913168433152, 435991257271370763778916352

Offset: 0

Paul D. Hanna, Dec 18 2013

nonn

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 1968*x^4/4! + 39488*x^5/5! +...
Related expansions.
A(x)^2 = 1 + 4*x + 32*x^2/2! + 400*x^3/3! + 6848*x^4/4! + 149056*x^5/5! +...
A(x)^4 = 1 + 8*x + 96*x^2/2! + 1568*x^3/3! + 32640*x^4/4! + 828032*x^5/5! +...
The logarithm of e.g.f. A(x) begins:
log(A(x)) = 2*x + 8*x^2/2! + 72*x^3/3! + 992*x^4/4! + 18336*x^5/5! +...
and equals Integral A(x) + A(x)^3 dx, where
A(x)^3 = 1 + 6*x + 60*x^2/2! + 864*x^3/3! + 16368*x^4/4! + 385344*x^5/5! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..350

Cf. A112487, A124214.

CoefficientList[Exp[InverseSeries[Series[1-Exp[-x]-ArcTan[Tanh[x/2]], {x, 0, 20}], x]],x]*Range[0, 20]! (* Vaclav Kotesovec, Dec 20 2013 *)

/* Explicit formula: */
{a(n)=local(A,X=x+x^2*O(x^n));A=exp(serreverse(1-exp(-X) - atan(tanh(X/2))));n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

/* By definition: A'(x) = A(x)^2 + A(x)^4: */
{a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A^2+A^4+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

/* From: A(x) = exp( Integral A(x) + A(x)^3 dx ): */
{a(n)=local(A=1+x); for(i=1, n, A=exp(intformal(A+A^3)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))

E.g.f. A(x) satisfies:
(1) A(x) = exp( Integral A(x) + A(x)^3 dx ) with A(0)=1.
(2) A(x) = (1 + B(x))/(1 - B(x)) where B(x) = tan(1-x - 1/A(x)).
(3) log(A(x)) = Series_Reversion( 1-exp(-x) - atan(tanh(x/2)) ).
(4) A( 1-exp(-x) - atan(tanh(x/2)) ) = exp(x).
a(n) ~ n! / (GAMMA(1/3) * 3^(1/3) * n^(2/3) * (1-Pi/4)^(n+1/3)). - Vaclav Kotesovec, Jan 26 2014

A176785 Sequence with e.g.f. g(x) = -(1/2)sqrt(2exp(-2*x)-1) + 1/2.

Original entry on oeis.org

0, 1, 0, 4, 24, 256, 3360, 53824, 1016064, 22095616, 543966720, 14955833344, 454227400704, 15103031627776, 545668238868480, 21286707282264064, 891735287528914944, 39926103010743156736

Offset: 0

Karol A. Penson, Apr 26 2010

			a(4) = 24: The 24 plane increasing trees on 4 vertices are
............................................................
.........1(x4 colors).......1(x4 colors).......1(x4 colors).
......../|\................/|\................/|\...........
......./.|.\............../.|.\............../.|.\..........
......2..3..4............2..4..3............3..2..4.........
............................................................
.........1(x4 colors).......1(x4 colors).......1(x4 colors).
......../|\................/|\................/|\...........
......./.|.\............../.|.\............../.|.\..........
......3..4..2............4..2..3............4..3..2.........
............................................................

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.

Cf. A000629, A000984, A124212, A124214, A229558.

max = 17; g[x_] := -(1/2)*Sqrt[2*Exp[-2*x] - 1] + 1/2; CoefficientList[ Series[ g[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Oct 05 2011 *)

x='x+O('x^66); concat ([0], Vec( serlaplace( serreverse( -1/2*log(1-2*x+2*x^2) ) ) ) ) \\ Joerg Arndt, Mar 01 2014

The e.g.f. A(x) satisfies the autonomous differential equation
A' = (1-2*A+2*A^2)/(1-2*A) with A(0) = 0. The compositional inverse of the e.g.f. is -1/2*log(1-2*x+2*x^2).
a(n) = (-1)^(n-1)*D^(n-1)(1) evaluated at x = 1, where D denotes the operator g(x) -> d/dx((x+1/x)*g(x)).
Applying [Bergeron et al., Theorem 1] to the result x = int {t = 0..A(x)} 1/phi(t), where phi(t) = (1-2*t+2*t^2)/(1-2*t) = 1+2*t^2+4*t^3+8*t^4+... leads to the following combinatorial interpretation for this sequence: a(n) gives the number of plane increasing trees on n vertices with no vertices of outdegree 1 and where each vertex of outdegree k >= 2 can be colored in 2^(k-1) ways. An example is given below. - Peter Bala, Sep 06 2011
a(n) ~ 2^(n-3/2)*n^(n-1)/(exp(n)*(log(2))^(n-1/2)). - Vaclav Kotesovec, Jun 28 2013
a(n+1) = 1/sqrt(2) * Sum_{k >= 0} (1/8)^k*binomial(2*k,k)*(2*k - 1)^n = 1/sqrt(2)*Sum_{k >= 0} (-1/2)^k*binomial(-1/2,k)*(2*k - 1)^n = Sum_{k = 0..n} Sum_{i = 0..k} (-1)^(k-i)/4^k* binomial(2*k,k)*binomial(k,i)*(2*i - 1)^n. Cf. A124212, A124214 and A229558. - Peter Bala, Aug 30 2016

A124215 E.g.f.: exp(exp(x)/(2-exp(3*x))^(1/3)-1).

Original entry on oeis.org

1, 2, 14, 166, 2742, 57734, 1475078, 44279414, 1526285814, 59383786278, 2573408251366, 122892818420310, 6411057873364822, 362707216829623046, 22117095828202632262, 1445932369337590029622, 100884144820332079749302, 7481824780607157773071590

Offset: 0

Karol A. Penson, Oct 19 2006

nonn

Exponential transform of A124214.

Cf.: A124214.

a(n) ~ 3^(n + 1/8) * exp(4*n^(1/4) / (3^(3/4) * log(2)^(1/4)) - n - 1) * n^(n - 3/8) / (2*log(2)^(n + 1/8)). - Vaclav Kotesovec, Jun 26 2022

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A124212 Expansion of e.g.f. exp(x)/sqrt(2-exp(2*x)).

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A229558 E.g.f.: exp(x) / (2 - exp(4*x))^(1/4).

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A228608 E.g.f. A(x) satisfies: A'(x) = A(x)^2 + A(x)^4.

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A176785 Sequence with e.g.f. g(x) = -(1/2)*sqrt(2*exp(-2*x)-1) + 1/2.

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A124215 E.g.f.: exp(exp(x)/(2-exp(3*x))^(1/3)-1).

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A176785 Sequence with e.g.f. g(x) = -(1/2)sqrt(2exp(-2*x)-1) + 1/2.