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.

A143139 E.g.f.: A(x) = exp(x + A(x)^2) - 1.

Original entry on oeis.org

1, 3, 25, 351, 6901, 174483, 5392465, 196967991, 8301682141, 396555037803, 21171512707225, 1249311005445231, 80742309245690821, 5672134436846492163, 430345858647623635105, 35069095795843414698471, 3054896437732455928549741, 283283784773408059496473563
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2008

Keywords

Examples

			A(x) = x + 3*x^2/2! + 25*x^3/3! + 351*x^4/4! + 6901*x^5/5! + ...
where A(log(1+x) - x^2) = x.
Log(1 + A(x)) = x + A(x)^2 = G(x) = g.f. of A143138:
G(x) = x + 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ...
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ...

Crossrefs

Cf. A143138.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[Log[1+x]-x^2, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Dec 28 2013 *)
  • Maxima
    a(n):=sum((n+k-1)!*sum((-1)^(j)/(k-j)!*sum(((-1)^l*stirling1(n-2*l+j-1,j-l))/(l!*(n-2*l+j-1)!),l,0,min(j,(n+j-1)/2)),j,0,k),k,0,n-1); /* Vladimir Kruchinin, Feb 17 2012 */
  • PARI
    {a(n)=local(A=x+O(x^n));for(i=0,n,A=exp(x+A^2)-1);n!*polcoeff(A,n)}
  • PARI
    {a(n)=n!*polcoeff(exp(serreverse(x-(exp(x+x*O(x^n))-1)^2))-1,n)}

Formula

E.g.f.: A(x) = Series_Reversion( log(1+x) - x^2 ).
E.g.f. derivative: A'(x) = (1 + A(x))/(1 - 2*A(x) - 2*A(x)^2 ).
a(n) = sum(k=0..n-1, (n+k-1)!*sum(j=0..k, (-1)^(j)/(k-j)!*sum(l=0..min(j,(n+j-1)/2), ((-1)^l*stirling1(n-2*l+j-1,j-l))/(l!*(n-2*l+j-1)!)))). - Vladimir Kruchinin, Feb 17 2012
a(n) ~ sqrt(1+1/sqrt(3)) * 2^(n-3/2) * n^(n-1) / (exp(n) * (sqrt(3)-2-2*log(sqrt(3)-1))^(n-1/2)). - Vaclav Kotesovec, Dec 28 2013

A143154 E.g.f.: A(x) = x + log(1 - A(x))^2.

Original entry on oeis.org

1, 2, 18, 262, 5320, 138728, 4419156, 166319424, 7221397848, 355312006392, 19537581248592, 1187337791554176, 79025863405440432, 5716937001401316000, 446654003380859659488, 37480492611898380241248
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2008

Keywords

Comments

Radius of convergence is r = (-1 + 6*A(r) - A(r)^2)/4 = 0.172815973872...
where A(r) = 1 - exp((A(r)-1)/2) = 0.2965325775...

Examples

			A(x) = x + 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! + ...
-log(1 - A(x)) = G(x) = the g.f. of A143155:
G(x) = x + 3*x^2/2! + 26*x^3/3! + 376*x^4/4! + 7614*x^5/5! + ...
G(x)^2 = 2*x^2/2! + 18*x^3/3! + 262*x^4/4! + 5320*x^5/5! + ...
Related expansions:
A(x) = x + log(1-x)^2 + d/dx log(1-x)^4/2! + d^2/dx^2 log(1-x)^6/3! + d^3/dx^3 log(1-x)^8/4! + ...
log(A(x)/x) = log(1-x)^2/x + d/dx (log(1-x)^4/x)/2! + d^2/dx^2 (log(1-x)^6/x)/3! + d^3/dx^3 (log(1-x)^8/x)/4! + ...

Crossrefs

Cf. A143155, A143138, A202356.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x-Log[1-x]^2,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 24 2014 *)
  • Maxima
    a(n):=((n-1)!*sum(binomial(n+k-1,n-1)*sum((-1)^(n+j-1)*binomial(k,j)*sum((binomial(j,l)*(2*(j-l))!*stirling1(n-l+j-1,2*(j-l)))/(n-l+j-1)!,l,0,j),j,0,k),k,0,n-1)); /* Vladimir Kruchinin, Feb 07 2012 */
  • PARI
    {a(n)=local(A=x+O(x^n));for(i=0,n,A=x + log(1-A)^2);n!*polcoeff(A,n)}
  • PARI
    {a(n)=n!*polcoeff(serreverse(x-log(1-x+x*O(x^n))^2),n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, log(1-x+x*O(x^n))^(2*m)/m!)); n!*polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, log(1-x+x*O(x^n))^(2*m)/x/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f.: A(x) = Series_Reversion( x - log(1 - x)^2 ).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) log(1-x)^(2*n)/n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (log(1-x)^(2*n)/x)/n! ).
E.g.f. derivative: A'(x) = (1 - A(x))/(1 - A(x) + 2*log(1 - A(x))).
a(n) = ((n-1)!*sum(k=0..n-1, binomial(n+k-1,n-1)*sum(j=0..k, (-1)^(n+j-1)*binomial(k,j)*sum(l=0..j, (binomial(j,l)*(2*(j-l))!*stirling1(n-l+j-1,2*(j-l)))/(n-l+j-1)!)))), n>0. - Vladimir Kruchinin, Feb 07 2012
a(n) ~ c*sqrt(4/(1+c)-2-2*c) * n^(n-1) / (exp(n) * (1-c*(2+c))^n), where c = LambertW(1/2) = 0.35173371124919... (see A202356). - Vaclav Kotesovec, Jan 24 2014

A277404 E.g.f. A(x) satisfies: A( x - (exp(x) - 1)^2 ) = x + (exp(x) - 1)^2.

Original entry on oeis.org

1, 4, 36, 508, 10020, 253804, 7853076, 287078908, 12106864260, 578586544204, 30901130685876, 1823983173981148, 117911755067635620, 8284976875099852204, 628692318063511556436, 51240154266491883376828, 4464155216699369664399300, 414013560595951627772296204, 40722939746084736801890208756
Offset: 1

Views

Author

Paul D. Hanna, Oct 15 2016

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 36*x^3/3! + 508*x^4/4! + 10020*x^5/5! + 253804*x^6/6! + 7853076*x^7/7! + 287078908*x^8/8! + 12106864260*x^9/9! + 578586544204*x^10/10! +...
such that A( x - (exp(x) - 1)^2 ) = x + (exp(x) - 1)^2.

Crossrefs

Cf. A143138.

Programs

  • PARI
    {a(n) = n!*polcoeff( -x + 2*serreverse( x - (exp(x +x*O(x^n)) - 1)^2 ), n)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) = 2*A143138(n) for n>1.
E.g.f. A(x) satisfies:
(1) A(x) = x + 2 * (exp( (A(x) + x)/2 ) - 1)^2.
(2) A(x) = -x + 2 * Series_Reversion( x - (exp(x)-1)^2 ).
(3) A(x) = x + 2 * Sum_{n>=1} d^(n-1)/dx^(n-1) (exp(x)-1)^(2*n) / n!.
(4) A( log(1+x) - x^2 ) = log(1+x) + x^2.
Showing 1-3 of 3 results.

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