OFFSET
0,3
FORMULA
E.g.f.: 1/(1 - 2*Series_Reversion( Integral (1 - 4*x^2)^2 dx ))^(1/2).
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 31*x^3/3! + 297*x^4/4! + 5521*x^5/5! +...
Related series.
A(x)^7 = 1 + 7*x + 63*x^2/2! + 805*x^3/3! + 13041*x^4/4! + 261247*x^5/5! +...
Note that 1 - 1/A(x)^2 is an odd function:
1 - 1/A(x)^2 = 2*x + 32*x^3/3! + 4352*x^5/5! + 1605632*x^7/7! +...
where Series_Reversion((1 - 1/A(x)^2)/2) = Integral (1-4*x^2)^2 dx.
PROG
(PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^7*subst(A^4, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=1); A=1/(1-2*serreverse(intformal((1-4*x^2 +x*O(x^n))^(4/2))))^(1/2); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 09 2014
STATUS
approved