OFFSET
1,3
COMMENTS
At what positions n is a(n) odd?
Compare g.f. to: C(x - C(x) + C(x)^2) = 0, trivial when C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..512
FORMULA
G.f. A(x) satisfies: x - A(x) + A(x)^2 = Ai(-x^4) where Ai( A(x) ) = x.
a(n) ~ c * d^n / n^(3/2), where d = 4.05999022767846206402248334679744980701174... and c = 0.14415462031792796731396571657... - Vaclav Kotesovec, Aug 28 2017
EXAMPLE
G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 16*x^5 + 48*x^6 + 152*x^7 + 501*x^8 + 1690*x^9 + 5822*x^10 + 20388*x^11 + 72360*x^12 + 259688*x^13 + 940792*x^14 + 3435904*x^15 + 12636554*x^16 + 46760376*x^17 + 173971252*x^18 + 650380288*x^19 + 2441905192*x^20 + 9203979808*x^21 +...
where A(x - A(x) + A(x)^2) = -x^4.
RELATED SERIES.
Define Ai(x) such that Ai(A(x)) = x, where Ai(x) begins:
Ai(x) = x - x^2 - x^4 + 4*x^5 - 6*x^6 + 8*x^7 - 30*x^8 + 92*x^9 - 190*x^10 + 428*x^11 - 1276*x^12 + 3524*x^13 - 8572*x^14 + 22120*x^15 - 62215*x^16 + 169464*x^17 - 444860*x^18 + 1202364*x^19 - 3340582*x^20 + 9167812*x^21 - 24936852*x^22 + 68746520*x^23 - 191319986*x^24 + 530404940*x^25 +...
then x - A(x) + A(x)^2 = Ai(-x^4),
and Ai(x) - Ai( -Ai(x)^4 ) = x - x^2.
PROG
(PARI) {a(n) = my(A=x, V=[1, 1, 2, 6]); for(i=1, n, V=concat(V, 0); A=x*Ser(V); V[#V]=Vec(subst(A, x, x - A + A^2))[#V-3]); V[n]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 20 2017
STATUS
approved