A058562 -id:A058562 - cp's OEIS frontend

A058575 The sequence S defined in A058562.

0, 0, 1, 7, 81, 1311, 27273, 693351, 20830113, 722035503, 28364200569, 1245313236663, 60429110073489, 3211572892464639, 185523138537151977, 11574425593731913479, 775591270444009771137, 55555665263291738684367, 4236182199641241492147801

Offset: 0

N. J. A. Sloane, Dec 26 2000

nonn

Cf. A058562.

spec := [ S, {N=Union(Z,S,P,Q), S=Set(Union(Z,P,Q),card>=2), P=Set(Union(Z,S,Q),card>=2), Q=Set(Union(Z,S,P),card>=2)}, labeled ]; [seq(combstruct[count](spec,size=n), n=0..40)]; # N=A058562, S=A058575
spec:=[S,{S=Set(Union(Z,S,S),card>=2)},labeled];[seq(combstruct[count](spec,size=n),n=0..20)]; # Vladeta Jovovic, Jun 25 2007

max = 18; se = Series[ -1/2*ProductLog[ -2/3*Exp[-2/3 + 1/3*x]] - 1/3 - x/3 , {x, 0, max}]; Join[{0, 0}, (CoefficientList[se, x] // DeleteCases[#, 0] &) ]* Range[0, max]! (* Jean-François Alcover, Jun 24 2013, after Vladeta Jovovic *)

E.g.f.: -(1/2)*LambertW(-2/3*exp(-2/3 + 1/3*x)) - 1/3 - x/3. - Vladeta Jovovic, Jun 25 2007

a(n) ~ sqrt(log(3/2)-1/3) * n^(n-1) / (2 * exp(n) * (log(27/8)-1)^n). - Vaclav Kotesovec, Jul 09 2013

A234294 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^4 dx.

Original entry on oeis.org

1, 1, 4, 40, 664, 15424, 460576, 16808320, 724904896, 36072438016, 2034328297984, 128223244372480, 8932539799788544, 681536817951791104, 56521548341146402816, 5062454448656689500160, 487013865350356256137216, 50082306316236214342844416, 5482502331779770770018893824

Offset: 0

Paul D. Hanna, Dec 24 2013

nonn

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 40*x^3/3! + 664*x^4/4! + 15424*x^5/5! +...
where A(4*log(1+x) - 3*x) = 1+x.
Related series:
A(x)^4 = 1 + 4*x + 28*x^2/2! + 328*x^3/3! + 5752*x^4/4! + 137056*x^5/5! +...
1/A(x)^4 = 1 - 4*x + 4*x^2/2! - 40*x^3/3! - 536*x^4/4! - 13216*x^5/5! +...
(d/dx Series_Reversion(Integral 1/A(x)^4 dx))^(1/4) begins:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 +...+ A002293(n)*x^n +...
where G(x) = 1 + x*G(x)^4.

Cf. A006351, A058562, A234295.

CoefficientList[1 + InverseSeries[Series[4*Log[1+x]-3*x, {x, 0, 20}], x],x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)

{a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^4+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

{a(n)=local(A=1, X=x+x^2*O(x^n)); A=1+serreverse(4*log(1+X) - 3*X); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* O.g.f. continued fraction: */
{a(n)=local(CF=1+x*O(x)); for(k=0, n, CF=1-(n-k+1)*x-3*(n-k+1)*x/CF); polcoeff(1+x/CF, n, x)}
for(n=0, 25, print1(a(n), ", "))

E.g.f.: 1 + Series_Reversion( 4*log(1+x) - 3*x ).
E.g.f.: -4/3*LambertW(-3/4*exp((x-3)/4)).
E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x)^4 dx ) )^(1/4), where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
E.g.f.: 1 / sqrt( d/dx Series_Reversion( Integral (1+2*x*C(2*x))^2/(1+x)^2 dx ) ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108.
O.g.f.: 1 + x/(1-x - 3*x/(1-2*x - 3*2*x/(1-3*x - 3*3*x/(1-4*x - 3*4*x/(1-...))))), a continued fraction.
a(n) ~ 2 * n^(n-1) / (3 * exp(n) * (8*log(2)-4*log(3)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013

A095839 a(n) = ((2n)!/(n!2^(n-1)))Integral_{x=1/2..1} (sqrt(1-x^2)/x)^(2n) dx.

Original entry on oeis.org

1, 1, 5, 51, 807, 17445, 479565, 16019955, 630301455, 28552506885, 1463744449125, 83780913568275, 5296205435649975, 366478026602012325, 27552067849812030525, 2236327624673777509875, 194908916445067162713375, 18154937081288124469477125

Offset: 0

Al Hakanson (Hawkuu(AT)excite.com), Jun 08 2004

nonn

From Paul Hanna, Dec 22 2013: (Start)
E.g.f. satisfies: A(x) = 1 + A(x)^3 * Integral 1/A(x) dx. Compare to: G(x) = 1 + G(x)^3 * Integral 1/G(x)^3 dx, where G(x)-1 is the e.g.f. of A058562, the number of 3-way series-parallel networks with n labeled edges.
A(x)^3 = 1 + 3*x + 21*x^2/2! + 249*x^3/3! + 4275*x^4/4! + 97155*x^5/5! +...
Integral 1/A(x) dx = x - x^2/2! - 3*x^3/3! - 27*x^4/4! - 405*x^5/5! - 8505*x^6/6! +...
Series_Reversion(Integral 1/A(x) dx) = x + x^2/2 + 3*x^3/3 + 12*x^4/4 + 55*x^5/5 + 273*x^6/6 + 1428*x^7/7 +...+ A001764(n-1)*x^n/n +...
E.g.f.: 1 + Series_Reversion( (x - x^2/2) / (1+x)^2 ). (End)

Robert Israel, Table of n, a(n) for n = 0..347

Cf. A058562, A234289, A234291, A001764, A000108.

A095839 := proc(n)
    local k;
    (4^k-2)/2/(2*k-1) ;
    add(%*(-1)^k*binomial(n,k),k=0..n) ;
    %*(-1)^n*(2*n)!/n!/2^(n-1) ;
end proc: # R. J. Mathar, Feb 13 2014

f[n_] := Numerator[ Integrate[(Sqrt[1 - x^2]/x)^(2n), {x, 1/2, 1}]*(2n)!/(n!2^(n + 1)!)]; Table[ f[n], {n, 0, 11}] (* Robert G. Wilson v *)
f[n_] := Numerator[2^(-2 - Gamma[2 + n])*3^(1 + n)*(2*n)!* Hypergeometric2F1Regularized[1, 1/2 + n, 2 + n, -3]]; Table[f[n], {n, 0, 11}] (* Eric W. Weisstein, Nov 19 2005 *)

{a(n)=local(A=(2-sqrt(1-6*x+x^2*O(x^n)))/(1+2*x)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 22 2013

D-finite with a(n) +(-4*n+9)*a(n-1) -6*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Feb 13 2014
E.g.f.: (2-sqrt(1-6*x))/(1+2*x). Recurrence follows from the d.e. (12*x^2+4*x-1)*y''+(30*x-1)*y'+6*y=0 satisfied by this. - Robert Israel, May 08 2018
a(n) ~ 2^(n-5/2) * 3^(n+1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Dec 27 2013

a(8)-a(11) from Robert G. Wilson v, Nov 18 2005

Definition corrected by Robert Israel, May 08 2018

A201465 E.g.f. satisfies: A(x) = (x + 2*exp(A(x)) - 2)/3.

Original entry on oeis.org

1, 2, 14, 162, 2622, 54546, 1386702, 41660226, 1444071006, 56728401138, 2490626473326, 120858220146978, 6423145784929278, 371046277074303954, 23148851187463826958, 1551182540888019542274, 111111330526583477368734, 8472364399282482984295602, 685178683361064789536947374

Offset: 1

Paul D. Hanna, Dec 01 2011

nonn

			E.g.f.: A(x) = x + 2*x^2 + 14*x^3/3! + 162*x^4/4! + 2622*x^5/5! + 54546*x^6/6! +...
The exponential of the e.g.f. begins:
exp(A(x)) = 1 + x + 3*x^2/2! + 21*x^3/3! + 243*x^4/4! + 3933*x^5/5! + 81819*x^6/6! +...
where x = 2 + 3*A(x) - 2*exp(A(x)).
...
O.g.f.: G(x) = 1 + 2*x + 14*x^2 + 162*x^3 + 2622*x^4 + 54546*x^5 +...
where
G(x) = 1/3 + 2/(3*(3-x)) + 2^2/(3*(3-x)*(3-2*x)) + 2^3/(3*(3-x)*(3-2*x)*(3-3*x)) + 2^4/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)) + 2^5/(3*(3-x)*(3-2*x)*(3-3*x)*(3-4*x)*(3-5*x)) +...

Cf. variants: A000311, A201466. A058562.

Rest[CoefficientList[1 + InverseSeries[Series[2 + 3*x - 2*Exp[x], {x, 0, 20}], x],x]* Range[0, 20]!] (* Vaclav Kotesovec, Dec 26 2013 *)

a(n):=(sum((n+k-1)!*sum(1/(k-j)!*sum((3^i*(-1)^(i)*2^(j-i)*stirling2(n+j-i-1,j-i))/(i!*(n+j-i-1)!),i,0,j),j,0,k),k,0,n-1)); /* Vladimir Kruchinin, Feb 04 2012 */

{a(n)=n!*polcoeff(serreverse(2+3*x - 2*exp(x+x^2*O(x^n))),n)}
for(n=0, 25, print1(round(A[n+1]), ", "))

\p100 \\ set precision
{A=Vec(sum(n=0, 600, 1.*2^n/prod(k=0, n, 3 - k*x + O(x^31))))}
for(n=0, 25, print1(round(A[n+1]), ", ")) \\ Paul D. Hanna, Oct 27 2014

E.g.f. A(x) satisfies: x = A( 2 + 3*x - 2*exp(x) ).
a(n)=(sum(k=0..n-1, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(i=0..j, (3^i*(-1)^(i)*2^(j-i)*stirling2(n+j-i-1,j-i))/(i!*(n+j-i-1)!))))), n>0. [From Vladimir Kruchinin, Feb 04 2012]
exp(A(x))-1 is the compositional inverse of 3*log(1+x)-2*x and is the e.g.f. of A058562. - Peter Bala, Jul 12 2012
G.f.: 1/Q(0), where Q(k)= 1 - k*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
E.g.f.: (x-2)/3 - LambertW(-2/3*exp((x-2)/3)). - Vaclav Kotesovec, Dec 26 2013
a(n) ~ n^(n-1) / (sqrt(3) * exp(n) * (3*log(3)-3*log(2)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013
O.g.f.: Sum_{n>=0} 2^n / Product_{k=0..n} (3 - k*x). - Paul D. Hanna, Oct 27 2014

A234290 Duplicate of A095839.

Original entry on oeis.org

1, 1, 5, 51, 807, 17445, 479565, 16019955, 630301455, 28552506885, 1463744449125, 83780913568275, 5296205435649975, 366478026602012325, 27552067849812030525, 2236327624673777509875, 194908916445067162713375, 18154937081288124469477125, 1799824448875247911270279125

Offset: 0

Paul D. Hanna, Dec 22 2013

dead

Compare to: G(x) = 1 + G(x)^3 * Integral 1/G(x)^3 dx, where G(x)-1 is the e.g.f. of A058562, the number of 3-way series-parallel networks with n labeled edges.

Is this sequence the same as A095839?

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 51*x^3/3! + 807*x^4/4! + 17445*x^5/5! +...
where A(x)^3 = 1 + 3*x + 21*x^2/2! + 249*x^3/3! + 4275*x^4/4! + 97155*x^5/5! +...
Integral 1/A(x) dx = x - x^2/2! - 3*x^3/3! - 27*x^4/4! - 405*x^5/5! - 8505*x^6/6! +...
Further,
Series_Reversion(Integral 1/A(x) dx) = x + x^2/2 + 3*x^3/3 + 12*x^4/4 + 55*x^5/5 + 273*x^6/6 + 1428*x^7/7 +...+ A001764(n-1)*x^n/n +...
where A001764(n) = binomial(3*n,n)/(2*n+1).

Cf. A058562, A234289, A234291, A001764, A000108.

CoefficientList[Series[(2-Sqrt[1-6*x])/(1+2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 27 2013 *)

{a(n)=local(A=1+x);for(i=1,n,A=1+A^3*intformal(1/(A+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))

/* From formula using g.f. of A001764 G(x) = 1 + x*G(x)^3: */
{a(n)=local(G=sum(m=0,n,binomial(3*m,m)/(2*m+1)*x^m)+x*O(x^n),A=1);A=1/deriv(serreverse(intformal(G))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* Explicit formula: 1/(1 - x*C(3*x/2)), C(x) = 1 + x*C(x)^2 */
{a(n)=local(A=(2-sqrt(1-6*x+x^2*O(x^n)))/(1+2*x)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* From formula: 1 + Series_Reversion((x - x^2/2)/(1+x)^2): */
{a(n)=local(A=1,X=x+x^2*O(x^n));A=1+serreverse((X-X^2/2)/(1+X)^2); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x) dx ) ) where G(x) = 1 + x*G(x)^3 = Sum_{n>=0} A001764(n)*x^n is the g.f. of A001764.
E.g.f.: (2 - sqrt(1-6*x)) / (1+2*x) = 1/(1 - x*C(3*x/2)), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers (A000108).
E.g.f.: 1 + Series_Reversion( (x - x^2/2) / (1+x)^2 ).
a(n) ~ 2^(n-5/2) * 3^(n+1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Dec 27 2013
D-finite with recurrence a(n) +(-4*n+9)*a(n-1) -6*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

A234291 E.g.f. satisfies: A(x) = 1 + A(x)^3 * Integral 1/A(x)^2 dx.

Original entry on oeis.org

1, 1, 4, 34, 460, 8608, 206152, 6020992, 207574240, 8251015264, 371527296256, 18691127602816, 1039066330203520, 63253339835514112, 4184830238170091008, 298985971407749744128, 22941517126450315985920, 1881603821848104123344896, 164271703613261014954276864

Offset: 0

Paul D. Hanna, Dec 22 2013

nonn

Compare to: G(x) = 1 + G(x)^3 * Integral 1/G(x)^3 dx, where G(x)-1 is the e.g.f. of A058562, the number of 3-way series-parallel networks with n labeled edges.

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 460*x^4/4! + 8608*x^5/5! +...
where A(x)^3 = 1 + 3*x + 18*x^2/2! + 180*x^3/3! + 2628*x^4/4! + 51264*x^5/5! +...
Integral 1/A(x)^2 dx = x - 2*x^2/2! - 2*x^3/3! - 20*x^4/4! - 272*x^5/5! - 5096*x^6/6! +...
Further,
Series_Reversion(Integral 1/A(x)^2 dx) = x + 2*x^2/2 + 7*x^3/3 + 30*x^4/4 + 143*x^5/5 + 728*x^6/6 + 3876*x^7/7 +...+ A006013(n-1)*x^n/n +...
where A006013(n) = binomial(3*n+1,n)/(n+1).

Cf. A001764, A006013, A058562, A234290, A234289.

seq(n! * coeff(series(-3/(2*LambertW(-1,-3/2*exp((x-3)/2))), x, n+1), x, n), n = 0..10) # Vaclav Kotesovec, Dec 27 2013

CoefficientList[1 + InverseSeries[Series[3*x/(1+x) - 2*Log[1+x], {x, 0, 20}], x],x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 27 2013 *)

{a(n)=local(A=1+x); for(i=1, n, A=1+A^3*intformal(1/(A^2+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* Explicit formula using g.f. of A001764, G(x) = 1 + x*G(x)^3: */
{a(n)=local(G=sum(m=0, n, binomial(3*m, m)/(2*m+1)*x^m)+x*O(x^n), A=1); A=1/sqrt(deriv(serreverse(intformal(G^2)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* Explicit formula: 1 + Series_Reversion(3*x/(1+x) - 2*log(1+x)): */
{a(n)=local(A=1, X=x+x^2*O(x^n)); A=1+serreverse(3*X/(1+X)-2*log(1+X)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

E.g.f.: 1 / sqrt( d/dx Series_Reversion( Integral G(x)^2 dx ) ) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764, and G(x)^2 is the g.f. of A006013.
E.g.f.: 1 + Series_Reversion( 3*x/(1+x) - 2*log(1+x) ).
E.g.f.: -3/(2*LambertW(-1,-3/2*exp((x-3)/2))). - Vaclav Kotesovec, Dec 27 2013
a(n) ~ 3*sqrt(2) * n^(n-1) / (4*exp(n) * (1+2*log(2)-2*log(3))^(n-1/2)). - Vaclav Kotesovec, Dec 27 2013

A234295 E.g.f. satisfies: A(x) = 1 + A(x)^5 * Integral 1/A(x)^5 dx.

Original entry on oeis.org

1, 1, 5, 65, 1405, 42505, 1653125, 78578225, 4414067725, 286099718425, 21015972365525, 1725374840578625, 156560122048892125, 15559151967183795625, 1680744724811088153125, 196083244062052339084625, 24570430118524659881918125, 3291153805391398126661325625

Offset: 0

Paul D. Hanna, Dec 25 2013

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 65*x^3/3! + 1405*x^4/4! + 42505*x^5/5! +...
where A(5*log(1+x) - 4*x) = 1+x.
Related series:
A(x)^5 = 1 + 5*x + 45*x^2/2! + 685*x^3/3! + 15645*x^4/4! + 485645*x^5/5! +...
1/A(x)^5 = 1 - 5*x + 5*x^2/2! - 85*x^3/3! - 1595*x^4/4! - 50645*x^5/5! +...
(d/dx Series_Reversion(Integral 1/A(x)^5 dx))^(1/5) begins:
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 +...+ A002294(n)*x^n +...
where G(x) = 1 + x*G(x)^5.

Cf. A006351, A058562, A234294.

CoefficientList[1 + InverseSeries[Series[5*Log[1+x]-4*x, {x, 0, 20}], x],x]* Range[0, 20]! (* Vaclav Kotesovec, Dec 26 2013 *)

{a(n)=local(A=1+x); for(i=1, n, A=1+A^5*intformal(1/(A^5+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

{a(n)=local(A=1,X=x+x^2*O(x^n)); A=1+serreverse(5*log(1+X) - 4*X); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* O.g.f. continued fraction: */
{a(n)=local(CF=1+x*O(x)); for(k=0, n, CF=1-(n-k+1)*x-4*(n-k+1)*x/CF); polcoeff(1+x/CF, n, x)}
for(n=0, 25, print1(a(n), ", "))

E.g.f.: 1 + Series_Reversion( 5*log(1+x) - 4*x ).
E.g.f.: -5/4*LambertW(-4/5*exp((x-4)/5)).
E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x)^5 dx ) )^(1/5), where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
O.g.f.: 1 + x/(1-x - 4*x/(1-2*x - 4*2*x/(1-3*x - 4*3*x/(1-4*x - 4*4*x/(1-...))))), a continued fraction.
a(n) ~ sqrt(5) * n^(n-1) / (4*exp(n)*(5*log(5)-10*log(2)-1)^(n-1/2)). - Vaclav Kotesovec, Dec 26 2013

cp's OEIS Frontend

A058575 The sequence S defined in A058562.

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A234294 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x)^4 dx.

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A095839 a(n) = ((2*n)!/(n!*2^(n-1)))*Integral_{x=1/2..1} (sqrt(1-x^2)/x)^(2*n) dx.

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A201465 E.g.f. satisfies: A(x) = (x + 2*exp(A(x)) - 2)/3.

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Maxima

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A234290 Duplicate of A095839.

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A234291 E.g.f. satisfies: A(x) = 1 + A(x)^3 * Integral 1/A(x)^2 dx.

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A234295 E.g.f. satisfies: A(x) = 1 + A(x)^5 * Integral 1/A(x)^5 dx.

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A095839 a(n) = ((2n)!/(n!2^(n-1)))Integral_{x=1/2..1} (sqrt(1-x^2)/x)^(2n) dx.