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.

A227852 G.f. A(x) satisfies: A(x) = Series_Reversion( x - (A(x)^2 + A(-x)^2)/2 ).

Original entry on oeis.org

1, 1, 2, 10, 44, 294, 1728, 13389, 93130, 796620, 6235288, 57551130, 493813936, 4857378920, 44989814920, 468103507718, 4633862094852, 50749496457992, 533271010341720, 6126256486912776, 67990630238066888, 817168635245112432, 9541543704324657008, 119719059789052412360
Offset: 1

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Author

Paul D. Hanna, Oct 31 2013

Keywords

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 10*x^4 + 44*x^5 + 294*x^6 + 1728*x^7 +...
The series reversion of A(x), G(x) where A(G(x)) = x, begins:
G(x) = x - x^2 - 5*x^4 - 112*x^6 - 4320*x^8 - 227766*x^10 - 14942616*x^12 - 1162657840*x^14 +...+ (-1)^n * A263531(n)*x^(2*n) +...
and can be formed from a bisection of A(x)^2:
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 24*x^5 + 112*x^6 + 716*x^7 + 4320*x^8 + 32290*x^9 + 227766*x^10 + 1893488*x^11 + 14942616*x^12 + 134816212*x^13 + 1162657840*x^14 +...
The related g.f. of A263531, F(x) = -(A(I*x)^2 + A(-I*x)^2)/2, 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 - 5*x^4 + 112*x^6 - 4320*x^8 + 227766*x^10 - 14942616*x^12 +...

Links

Crossrefs

Cf. A263531, A213591, A141202.

Programs

  • PARI
    {a(n)=local(A=x);for(i=1,n,A=serreverse(x-(A^2+subst(A^2,x,-x +x*O(x^n)))/2));polcoeff(A,n)}
for(n=1,25,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^2+subst(A,x,-x)^2)^m/2^m/m!))+x*O(x^n)); polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = x + (A(A(x))^2 + A(-A(x))^2)/2.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (A(x)^2 + A(-x)^2)^n / (n!*2^n).
(3) (A(I*x)^2 + A(-I*x)^2)/2 = -F(x), where F(x) is the g.f. of A263531 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.

A263530 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, -3, 52, -1596, 68174, -3679964, 238949640, -18133397519, 1578639190316, -155623090726884, 17203681850199360, -2116171636238243028, 287762930191296817296, -43014624174283817327952, 7032470676704382424751408, -1251802142595596587066746328, 241602713767787669715442097616, -50368862903110844612768593045136, 11303387910446267256159298807620472
Offset: 1

Views

Author

Paul D. Hanna, Oct 20 2015

Keywords

Examples

			G.f.: A(x) = x^2 - 3*x^4 + 52*x^6 - 1596*x^8 + 68174*x^10 - 3679964*x^12 + 238949640*x^14 - 18133397519*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 + 32*x^5 - 944*x^7 + 39366*x^9 - 2090576*x^11 + 134136792*x^13 - 10085875720*x^15 + 871536657504*x^17 +...+ (-1)^(n-1)*A141202(2*n-1)*x^(2*n-1) +...
C(x) = x^2 - 8*x^4 + 178*x^6 - 6255*x^8 + 293652*x^10 - 17085798*x^12 + 1182991528*x^14 - 95087538324*x^16 +...+ (-1)^(n-1)*A141202(2*n)*x^(2*n) +...
and
B(x)^2 = x^2 - 4*x^4 + 68*x^6 - 2016*x^8 + 83532*x^10 - 4399032*x^12 + 280046448*x^14 - 20916418480*x^16 + 1797498262020*x^18 +...
C(x)^2 = x^4 - 16*x^6 + 420*x^8 - 15358*x^10 + 719068*x^12 - 41096808*x^14 + 2783020961*x^16 - 218859071704*x^18 +...
Further
G(x) = -I*B(I*x) - C(I*x) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 178*x^6 + 944*x^7 + 6255*x^8 + 39366*x^9 + 293652*x^10 +...+ A141202(n)*x^n +...
where G(x + G(x)*G(-x)) = x.

Links

Crossrefs

Cf. A141202, A263531.

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 A141202, where G(x + G(x)*G(-x)) = x, and B(x) + I*C(x) = Series_Reversion(x - I*A(x)), then
(1) G(x)*G(-x) = 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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