OFFSET
0,4
COMMENTS
Limit n->infinity (|a(n)|/n!)^(1/n) = 7/8 * sqrt(11+5*sqrt(5)). - Vaclav Kotesovec, Nov 15 2014
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..360
FORMULA
E.g.f.: (1 + 2*Series_Reversion(G(x)))^(1/2), where G(x) = ((1+2*x)^(7/2) - (1-2*x)^(7/2))/14 = x + Sum_{n>=1} x^(2*n+1)/(2*n+1)! * Product_{k=0..n-1} (3-4*k)*(5-4*k).
E.g.f. A(x) satisfies:
(1) A(x)^2 + A(-x)^2 = 2.
(2) A(x)^7 - A(-x)^7 = 14*x.
(3) (A(x) - A(-x))/2 = Series_Reversion(x + 2*x^3 - 4*x^5 + 8*x^7/7).
(4) x = (A(x)^7 - (2 - A(x)^2)^(7/2))/14.
EXAMPLE
E.g.f.: A(x) = 1 + x - x^2/2! - 12*x^3/3! + 45*x^4/4! + 1920*x^5/5! - 12285*x^6/6! - 812160*x^7/7! + 7372665*x^8/8! + 675993600*x^9/9! - 7946069625*x^10/10! +...
Related expansions.
A(x)^2 = 1 + 2*x - 30*x^3/3! + 4530*x^5/5! - 1914750*x^7/7! + 1589710050*x^9/9! - 2183722897950*x^11/11! +...
A(x)^7 = 1 + 7*x + 35*x^2/2! - 1995*x^4/4! + 523215*x^6/6! - 314976375*x^8/8! + 339403095675*x^10/10! +...
EXPLICIT FORMULA.
Let G(x) = ((1+2*x)^(7/2) - (1-2*x)^(7/2))/14, which begins
G(x) = x + 15*x^3/3! - 15*x^5/5! - 225*x^7/7! - 14175*x^9/9! - 2027025*x^11/11! - 516891375*x^13/13! +...+ [Product_{k=0..n-1} (3-4*k)*(5-4*k)]*x^(2*n+1)/(2*n+1)! +...
then (A(x)^2 - 1)/2 = Series_Reversion(G(x)).
A series bisection may be expressed by the series reversion given by:
Series_Reversion(x + 2*x^3 - 4*x^5 + 8*x^7/7) = x - 12*x^3/3! + 1920*x^5/5! - 812160*x^7/7! + 675993600*x^9/9! +...
MATHEMATICA
CoefficientList[Sqrt[1 + 2*InverseSeries[Series[((1+2*x)^(7/2) - (1-2*x)^(7/2))/14, {x, 0, 20}], x]], x] * Range[0, 20]! (* Vaclav Kotesovec, Nov 15 2014 *)
PROG
(PARI) /* Explicit formula: */
{a(n)=local(A, X=x+x^2*O(x^n), G=((1+2*X)^(7/2) - (1-2*X)^(7/2))/14);
A=(1 + 2*serreverse(G))^(1/2); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Formula using series expansion: */
{a(n)=local(A, G=x + sum(m=1, n\2+1, x^(2*m+1)/(2*m+1)!*prod(k=0, m-1, (3-4*k)*(5-4*k)) +x^2*O(x^n)));
A=sqrt(1 + 2*serreverse(G)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Alternating zero coefficients in A(x)^2 and A(x)^7: */
{a(n)=local(A=[1, 1], E=1, M); for(i=1, n, A=concat(A, 0); M=#A;
E=sum(m=0, M-1, A[m+1]*x^m/m!)+x*O(x^M);
A[M]=if(M%2==0, -(M-1)!*Vec(E^7/7)[M], -(M-1)!*Vec(E^2/2)[M])); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 15 2014
STATUS
approved