OFFSET
1,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..300
FORMULA
G.f. A(x) satisfies:
(1) A(x) = Series_Reversion( x - Sum_{n>=1} 2*(n+2) * a(n) * x^(n+1)/(n+1) ).
(2) A(x) = x + Sum_{n>=1} 2*(n+2) * a(n) * A(x)^(n+1) / (n+1).
Let B(x) = Integral 2*A(x) dx, then
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (2*x*A(x) + B(x))^n / n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (2*x*A(x) + B(x))^n / (n!*x) ).
a(n) = (n+1)*A259611(n+1)/2 for n>=1.
EXAMPLE
G.f. A(x) = x + 3*x^2 + 26*x^3 + 320*x^4 + 4776*x^5 + 81018*x^6 +...
such that
Series_Reversion(A(x)) = x - 3*x^2 - 8*x^3 - 65*x^4 - 768*x^5 - 11144*x^6 - 185184*x^7 - 3398256*x^8 - 67395940*x^9 +...+ -2*(n+2)/(n+1)*a(n)*x^(n+1) +...
Let B(x) = Integral 2*A(x) dx, an integer series that begins:
B(x) = x^2 + 2*x^3 + 13*x^4 + 128*x^5 + 1592*x^6 + 23148*x^7 + 377584*x^8 + 6739594*x^9 +...+ A259611(n)*x^n +...
then A(x - 2*x*A(x) - B(x)) = x.
Also,
A(x) = x + (2*x*A(x) + B(x)) + [d/dx (2*x*A(x) + B(x))^2]/2! + [d^2/dx^2 (2*x*A(x) + B(x))^3]/3! + [d^3/dx^3 (2*x*A(x) + B(x))^4]/4! + [d^4/dx^4 (2*x*A(x) + B(x))^5]/5! +...
Logarithmic series:
log(A(x)/x) = (2*x*A(x) + B(x))/x + [d/dx (2*x*A(x) + B(x))^2/x]/2! + [d^2/dx^2 (2*x*A(x) + B(x))^3/x]/3! + [d^3/dx^3 (2*x*A(x) + B(x))^4/x]/4! + [d^4/dx^4 (2*x*A(x) + B(x))^5/x]/5! +...
PROG
(PARI) {a(n)=local(A=x); for(i=0, n, A = serreverse(x - 2*x*A - intformal(2*A) +x*O(x^n))); 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, B=x^2); for(i=1, n, B=intformal(2*A); A = x + sum(m=1, n, Dx(m-1, (2*x*A + B)^m/m!)) +O(x^(n+1))); 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, B=x^2); for(i=1, n, B=intformal(2*A); A = x*exp(sum(m=1, n, Dx(m-1, (2*x*A + B)^m/(m!*x))) +O(x^(n+1)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 30 2015
STATUS
approved