OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = x + A(x/(1-x)) * A(x/(1+x)).
(2) A(x) = (A(x/(1+x)) - x/(1+x)) / A(x/(1+2*x)).
(3) A(x) = (A(x/(1-x)) - x/(1-x)) / A(x/(1-2*x)).
(4) A(x) = x + Sum_{n>=1} x/(1+n*x) * Product_{k=0..n-1} A( x/(1+(k-1)*x) ).
(5) A(x) = x + Sum_{n>=1} x/(1-n*x) * Product_{k=0..n-1} A( x/(1-(k-1)*x) ).
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 75*x^6 + 316*x^7 + 1450*x^8 + 7184*x^9 + 38261*x^10 + 217216*x^11 + 1308305*x^12 + ...
where A(x) = x + A(x/(1+x)) * A(x/(1-x)).
Also, A(x/(1-x)) = x/(1-x) + A(x)*x/(1-2*x) + A(x)*A(x/(1-x))*x/(1-3*x) + A(x)*A(x/(1-x))*A(x/(1-2*x))*x/(1-4*x) + A(x)*A(x/(1-x))*A(x/(1-2*x))*A(x/(1-3*x))*x/(1-5*x) + ...
RELATED SERIES.
A(x/(1-x)) = x + 2*x^2 + 5*x^3 + 16*x^4 + 61*x^5 + 261*x^6 + 1223*x^7 + 6197*x^8 + 33633*x^9 + 194217*x^10 + ...
where A(x/(1-x)) = x/(1-x) + A(x) * A(x/(1-2*x)).
A(x/(1+x)) = x + 0*x^2 + x^3 + 2*x^4 + 5*x^5 + 19*x^6 + 71*x^7 + 287*x^8 + 1345*x^9 + 6631*x^10 + 34907*x^11 + ...
where A(x/(1+x)) = x/(1+x) + A(x) * A(x/(1+2*x)).
PROG
(PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( x + subst(Ax, x, x/(1-x +x*O(x^#A)) )*subst(Ax, x, x/(1+x +x*O(x^#A)) ) - Ax, #A-1) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 14 2024
STATUS
approved