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.

A141202 G.f. satisfies: A(x + A(x)*A(-x)) = x.

Original entry on oeis.org

1, 1, 2, 8, 32, 178, 944, 6255, 39366, 293652, 2090576, 17085798, 134136792, 1182991528, 10085875720, 95087538324, 871536657504, 8727880568468, 85385942061016, 904071273001352, 9389429908430784, 104728235042891360, 1149676904405092704, 13467595558130095308, 155705728677310569008
Offset: 1

Views

Author

Paul D. Hanna, Jun 13 2008, Sep 05 2008

Keywords

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 178*x^6 + 944*x^7 +...
By definition, Series_Reversion(A(x)) = x + A(-x)*A(x) where
A(-x)*A(x) = -x^2 - 3*x^4 - 52*x^6 - 1596*x^8 - 68174*x^10 - 3679964*x^12 +...+ (-1)^n * A263530(n)*x^(2*n) +...
Consequently, A(x) = x - A(-A(x))*A(A(x)) where
-A(-A(x)) = x + 0*x^2 + 2*x^3 + x^4 + 30*x^5 + 38*x^6 + 852*x^7 +...
A(A(x)) = x + 2*x^2 + 6*x^3 + 27*x^4 + 134*x^5 + 786*x^6 + 4852*x^7 +...
The related g.f. of A263530, F(x) = A(-I*x)*A(I*x), satisfies: F(x) = B(x)^2 + C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1:
F(x) = x^2 - 3*x^4 + 52*x^6 - 1596*x^8 + 68174*x^10 - 3679964*x^12 +...

Links

Crossrefs

Cf. A263530, A227852.

Programs

  • Mathematica
    m = 26; A[_] = 0;
Do[A[x_] = x - A[-A[x]] A[A[x]] + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Oct 01 2019 *)
  • PARI
    {a(n)=local(A=x+x^2);for(i=0,n,A=serreverse(x+A*subst(A,x,-x+x*O(x^n)))) ;polcoeff(A,n)}
for(n=1,30,print1(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)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, (A*subst(-A, x, -x))^m/m!))+x*O(x^n)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = x - A(-A(x)) * A(A(x)).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (-A(x)*A(-x))^n / n!.
(3) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (-A(x)*A(-x))^n / (n!*x) ).
(4) A(-I*x) * A(I*x) = F(x), where F(x) is the g.f. of A263530 and satisfies: F(x) = B(x)^2 + C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1.

Extensions

Edited by N. J. A. Sloane, Sep 13 2008 at the suggestion of R. J. Mathar

A263531 G.f. A(x) satisfies: A(x) = B(x)^2 - C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*A(x)), where I^2 = -1.

Original entry on oeis.org

1, -5, 112, -4320, 227766, -14942616, 1162657840, -104338906529, 10609887976616, -1207797487940348, 152572977202977992, -21242819435887437760, 3241842130718219392320, -539712032454499745200600, 97612800729251959183577168, -19106581507633892101354812324, 4033513580481891302243479168168, -915408408852469072798058443048672
Offset: 1

Views

Author

Paul D. Hanna, Oct 20 2015

Keywords

Examples

			G.f.: A(x) = x^2 - 5*x^4 + 112*x^6 - 4320*x^8 + 227766*x^10 - 14942616*x^12 + 1162657840*x^14 - 104338906529*x^16 +...
such that A(x) = B(x)^2 - C(x)^2 and B(x) and C(x) are defined by
Series_Reversion(x - I*A(x)) = B(x) + I*C(x), where
B(x) = x - 2*x^3 + 44*x^5 - 1728*x^7 + 93130*x^9 - 6235288*x^11 + 493813936*x^13 - 44989814920*x^15 + 4633862094852*x^17 +...+ (-1)^(n-1)*A227852(2*n-1)*x^(2*n-1) +...
C(x) = x^2 - 10*x^4 + 294*x^6 - 13389*x^8 + 796620*x^10 - 57551130*x^12 + 4857378920*x^14 - 468103507718*x^16 +...+ (-1)^(n-1)*A227852(2*n)*x^(2*n)
and
B(x)^2 = x^2 - 4*x^4 + 92*x^6 - 3632*x^8 + 195108*x^10 - 12995160*x^12 + 1023750448*x^14 - 92825448208*x^16 + 9521361427980*x^18 +...
C(x)^2 = x^4 - 20*x^6 + 688*x^8 - 32658*x^10 + 1947456*x^12 - 138907392*x^14 + 11513458321*x^16 - 1088526548636*x^18 +...
Further
G(x) = -I*B(I*x) - C(I*x) = x + x^2 + 2*x^3 + 10*x^4 + 44*x^5 + 294*x^6 + 1728*x^7 + 13389*x^8 + 93130*x^9 + 796620*x^10 +...+ A227852(n)*x^n +...
where G( x - (G(x)^2 + G(-x)^2)/2 ) = x.

Links

Crossrefs

Cf. A227852, A263530.

Programs

  • PARI
    {a(n) = my(A=x^2, D); for(i=0,2*n, D=serreverse(x - I*A +O(x^(2*n+1))); A = real(D)^2 - imag(D)^2  ); polcoeff(A,2*n)}
for(n=1,20,print1(a(n),", "))
  • PARI
    /* Differential Series */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A = x^2 +O(x^(2*n+2))); for(i=1, 2*n, D = x + sum(m=1, 2*n, I^m*Dx(m-1, A^m/m!) +O(x^(2*n+2))); A = real(D)^2 - imag(D)^2 ); polcoeff(A, 2*n)}
for(n=1, 20, print1(a(n), ", "))

Formula

Let G(x) be the g.f. of A227852, where G( x - (G(x)^2 + G(-x)^2)/2 ) = x, and B(x) + I*C(x) = Series_Reversion(x - I*A(x)), then
(1) (G(x)^2 + G(-x)^2)/2 = -A(I*x).
(2) G(x + A(I*x)) = x.
(3) G(x) = x - A( I*G(x) ).
(4) G(x) = -I*B(I*x) - C(I*x), where A(x) = B(x)^2 - C(x)^2.
(5) B(x) + I*C(x) = x - Sum_{n>=1} d^(n-1)/dx^(n-1) I^n * A(x)^n/n!, where A(x) = B(x)^2 - C(x)^2.
Showing 1-2 of 2 results.

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