cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

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

Keywords

Comments

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)

Links

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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

Formula

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

Extensions

a(8)-a(11) from Robert G. Wilson v, Nov 18 2005
Definition corrected by Robert Israel, May 08 2018

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

Views

Author

Paul D. Hanna, Dec 22 2013

Keywords

Comments

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?

Examples

			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).

Crossrefs

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

Programs

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

Formula

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

Views

Author

Paul D. Hanna, Dec 22 2013

Keywords

Comments

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.

Examples

			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).

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {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), ", "))
  • PARI
    /* 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), ", "))
  • PARI
    /* 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), ", "))

Formula

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
Showing 1-3 of 3 results.

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