A291190 G.f. satisfies: A(x - A(x) + A(x)^2) = x^4.

1, 1, 2, 4, 12, 36, 112, 361, 1186, 3974, 13524, 46612, 162384, 570880, 2022800, 7216480, 25900036, 93449752, 338772408, 1233326352, 4507204720, 16528765376, 60805491392, 224335046602, 829851744732, 3077246265612, 11436732740472, 42593968518536, 158941264247584, 594169284671232, 2224933015422432, 8344687554060528, 31343475208937024, 117893400330845424, 444019302263216224

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 = 3.93460560538976027645396919840971895891402... and c = 0.137506207625998211308202134... - Vaclav Kotesovec, Aug 28 2017

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 12*x^5 + 36*x^6 + 112*x^7 + 361*x^8 + 1186*x^9 + 3974*x^10 + 13524*x^11 + 46612*x^12 + 162384*x^13 + 570880*x^14 + 2022800*x^15 + 7216480*x^16 + 25900036*x^17 + 93449752*x^18 + 338772408*x^19 + 1233326352*x^20 + 4507204720*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 - 28*x^8 + 92*x^9 - 146*x^10 - 36*x^11 + 968*x^12 - 3076*x^13 + 4628*x^14 + 3112*x^15 - 39947*x^16 + 119776*x^17 - 163020*x^18 - 205356*x^19 + 1800122*x^20 - 5042852*x^21 + 5978324*x^22 + 12502776*x^23 - 85355762*x^24 + 222312900*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, 4]); 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

Cf. A291189.

Sequence in context: A226022 A272463 A217699 * A273955 A054542 A214198

Adjacent sequences: A291187 A291188 A291189 * A291191 A291192 A291193

Keyword

nonn

Author

Paul D. Hanna, Aug 20 2017

Status

approved