OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
FORMULA
G.f. A(x) satisfies:
(1) A(x) = sqrt( Dx(A(x)) / (1 + x^2*Dx^2(A(x))) ), where Dx(F(x)) = d/dx x*F(x).
(2) A(x) = sqrt( (A(x) + x*A'(x)) / (1 + x^2*A(x) + 3*x^3*A'(x) + x^4*A''(x)) ).
(3) A(x) = sqrt( A'(x) / (1 + 2*x*A(x) + 4*x^2*A'(x) + x^3*A''(x)) ).
(4) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the g.f. of A245310.
(5) [x^n] (1-n + n*A(x)) * exp(-n*x*A(x)) = 0 for n>=1. - Paul D. Hanna, Jul 30 2018
a(n) = [x^n] G(x)^(n+1)/(n+1) for n>=0 where G(x) is the g.f. of A245310.
a(n) = [x^(n+2)] G(x)^(n+1)/(n+1)^2 for n>=0 where G(x) is the g.f. of A245310.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 50*x^5 + 182*x^6 + 707*x^7 +...
where A(x) = 1 / (1-x - x^2*A(x) - x^3*A'(x)).
Define Dx(A(x)) = d/dx x*A(x) = A(x) + x*A'(x):
Dx(A(x)) = 1 + 2*x + 6*x^2 + 20*x^3 + 75*x^4 + 300*x^5 + 1274*x^6 +...
so that Dx^2(A(x)) = d/dx x*(d/dx x*A(x)) = A(x) + 3*x*A'(x) + x^2*A''(x):
Dx^2(A(x)) = 1 + 4*x + 18*x^2 + 80*x^3 + 375*x^4 + 1800*x^5 + 8918*x^6 +...
then A(x)^2 = Dx(A(x)) / (1 + x^2*Dx^2(A(x))):
A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 44*x^4 + 150*x^5 + 549*x^6 + 2128*x^7 + 8673*x^8 + 36912*x^9 + 163288*x^10 +...
RELATED SERIES.
The g.f. of A245310 begins:
G(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 12*x^6 + 34*x^7 + 120*x^8 +...
where A(x) = G(x*A(x)) where G(x) = A(x/G(x)) and
a(n) = [x^n] G(x)^(n+1)/(n+1) = [x^(n+2)] G(x)^(n+1)/(n+1)^2 for n>=0.
PROG
(PARI) /* From A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^3*A(x)*A'(x): */
{a(n)=local(A=1+x); for(i=1, n, A = (1 + x^2*A^2 + x^3*A*A')/(1-x +x*O(x^n)));
polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x+2*x^2+sum(k=3, n-1, a(k)*x^k) +x^2*O(x^n));
if(n<3, polcoeff(A, n), -polcoeff( sqrt( deriv(x*A) / (1 + x^2*deriv(x*deriv(x*A)) +x^2*O(x^n)) ), n)/((n-1)/2))}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* From A(x) = G(x*A(x)) where G(x) is the g.f. of A245310: */
{a(n)=local(G=1+x+x^2); for(m=2, n,
G=G-x^(m+1)*(polcoeff(G^m+O(x^(m+2)), m+1)/m - polcoeff(G^m+O(x^m), m-1))+O(x^(n+3)));
polcoeff(1/x*serreverse(x/G+O(x^(n+3))), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 17 2014
STATUS
approved