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 1-2 of 2 results.

A234289 E.g.f. satisfies: A(x) = 1 + A(x)^2 * Integral 1/A(x) dx.

Original entry on oeis.org

1, 1, 3, 17, 147, 1729, 25827, 468593, 10012083, 246287521, 6856204803, 213102768977, 7315460977107, 274894137157249, 11223280473993507, 494715928976218673, 23416019742035332083, 1184519963466363339361, 63774753426394808946243, 3641219528568659379843857
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2013

Keywords

Comments

Compare to: G(x) = 1 + G(x)^2 * Integral 1/G(x)^2 dx, where G(x) is the e.g.f. of A006351, the number of series-parallel networks with n labeled edges.

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 147*x^4/4! + 1729*x^5/5! +...
where A(x)^2 = 1 + 2*x + 8*x^2/2! + 52*x^3/3! + 484*x^4/4! + 5948*x^5/5! +...
Integral 1/A(x) dx = x - x^2/2! - x^3/3! - 5*x^4/4! - 41*x^5/5! - 469*x^6/6! +...
Further,
Series_Reversion(Integral 1/A(x) dx) = x + x^2/2 + 2*x^3/3 + 5*x^4/4 + 14*x^5/5 + 42*x^6/6 + 132*x^7/7 +...+ A000108(n-1)*x^n/n +...
where A000108(n) = binomial(2*n,n)/(n+1).

Links

Crossrefs

Cf. A000108, A234290.

Programs

  • Maple
    seq(n! * coeff(series(-2/LambertW(-1,-2*exp(x-2)), x, n+1), x, n), n = 0..10) # Vaclav Kotesovec, Dec 27 2013
  • Mathematica
    CoefficientList[1 + InverseSeries[Series[2*x/(1+x) - 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^2*intformal(1/(A+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Explicit formula using Catalan function C(x) = 1 + x*C(x)^2: */
{a(n)=local(C=(1-sqrt(1-4*x+x^2*O(x^n)))/(2*x),A=1); A=1/deriv(serreverse(intformal(C))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* Explicit formula: 1 + Series_Reversion(2*x/(1+x) - log(1+x)): */
{a(n)=local(A=1,X=x+x^2*O(x^n)); A=1+serreverse(2*X/(1+X)-log(1+X)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f.: 1 / ( d/dx Series_Reversion( Integral C(x) dx ) ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108.
E.g.f.: 1 + Series_Reversion( 2*x/(1+x) - log(1+x) ).
E.g.f.: -2/LambertW(-1,-2*exp(x-2)). - Vaclav Kotesovec, Dec 27 2013
E.g.f.: A(x) = C( Integral 1/A(x) dx ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108. - Paul D. Hanna, May 23 2019
a(n) ~ 2 * n^(n-1) / (exp(n) * (1-log(2))^(n-1/2)). - Vaclav Kotesovec, Dec 27 2013

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 1-2 of 2 results.

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