OFFSET
1,3
COMMENTS
G.f. G(x) of A259270 satisfies: G(x) = Series_Reversion( x - 2*A(x)*G(x) ) such that G(x) = A'(x)/2, where A(x) = Sum_{n>=1} a(n)*x^(2*n) is the g.f. of this sequence.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..200
FORMULA
G.f. A(x) satisfies:
(1) A'(x) = 2*x + 2*Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n * A'(x)^n / n!.
(2) A'(x) = 2*x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n * A'(x)^n / (n!*x) ).
a(n) = A259270(n) / n.
a(n) == 1 (mod 2) iff n is a power of 2 (conjecture).
EXAMPLE
G.f.: A(x) = x^2 + x^4 + 6*x^6 + 61*x^8 + 818*x^10 + 13106*x^12 + 238636*x^14 +...
Let G(x) be the g.f. of A259270 such that
G(x) = A'(x)/2 = x + 2*x^3 + 18*x^5 + 244*x^7 + 4090*x^9 + 78636*x^11 + 1670452*x^13 + 38369256*x^15 +...+ A259270(n)*x^(2*n-1) +...
then G( x - 2*A(x)*G(x) ) = x.
Also,
A'(x)/2 = x + A(x)*A'(x) + [d/dx A(x)^2*A'(x)^2]/2! + [d^2/dx^2 A(x)^3*A'(x)^3]/3! + [d^3/dx^3 A(x)^4*A'(x)^4]/4! + [d^4/dx^4 A(x)^5*A'(x)^5]/5! +...
PROG
(PARI) {a(n)=local(A=x^2); for(i=1, n, A=intformal(2*serreverse(x - A*A' +O(x^(2*n))))); polcoeff(A, 2*n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {a(n)=local(A, G=x+x*O(x^n)); for(i=1, n, A=intformal(2*G); G = serreverse(x - 2*A*G +O(x^(2*n)))); polcoeff(A, 2*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^2); for(i=1, n, A = 2*intformal(x + sum(m=1, n+1, Dx(m-1, A^m*(A')^m/m!)) +O(x^(2*n+1)))); polcoeff(A, 2*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^2); for(i=1, n, A = 2*intformal(x*exp(sum(m=1, n, Dx(m-1, A^m*(A')^m/(m!*x))) +O(x^(2*n+1))))); polcoeff(A, 2*n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 29 2015
STATUS
approved