A384830 G.f. satisfies A(x) = A(x^3 - 3*x*A(x)^3) / A(x^2 - 2*x*A(x)^2).

1, -1, 4, -21, 110, -616, 3738, -23619, 152470, -1003776, 6726702, -45720504, 314307018, -2181641134, 15269811260, -107651952999, 763745165826, -5448656285938, 39063995033178, -281309141648214, 2033846965665156, -14757571862304042, 107431429198117338, -784411267743868602, 5743068864740600214

Offset: 1

Author

Paul D. Hanna, Jul 11 2025

Keywords

• sign

Comments

Compare to C(x) = C(x^3 + 3*x*C(x)^3) / C(x^2 + 2*x*C(x)^2), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).

Conjectures:

(C.1) a(n) is odd iff n = 2^k for k >= 0.

(C.2) a(n) == 1 (mod 3) iff n = 3^k for k >= 0.

(C.3) a(n) == 2 (mod 3) iff n = A038464(k)/2 for k >= 1, where A038464 lists sums of 2 distinct powers of 3.

Examples

			G.f.: A(x)  = x - x^2 + 4*x^3 - 21*x^4 + 110*x^5 - 616*x^6 + 3738*x^7 - 23619*x^8 + 152470*x^9 - 1003776*x^10 + 6726702*x^11 - 45720504*x^12 + 314307018*x^13 - 2181641134*x^14 + 15269811260*x^15 - 107651952999*x^16 +- ...
where A(x) = A(x^3 - 3*x*A(x)^3) / A(x^2 - 2*x*A(x)^2).
RELATED SERIES.
A(x)^2 = x^2 - 2*x^3 + 9*x^4 - 50*x^5 + 278*x^6 - 1620*x^7 + 10029*x^8 - 64262*x^9 + 420054*x^10 - 2793960*x^11 + 18874530*x^12 + ...
A(x)^3 = x^3 - 3*x^4 + 15*x^5 - 88*x^6 + 516*x^7 - 3123*x^8 + 19771*x^9 - 128748*x^10 + 853182*x^11 - 5739306*x^12 + ...
A(x^3 - 3*x*A(x)^3) = x^3 - 3*x^4 + 9*x^5 - 46*x^6 + 270*x^7 - 1575*x^8 + 9517*x^9 - 60228*x^10 + 391950*x^11 - 2595666*x^12 + ...
A(x^2 - 2*x*A(x)^2) = x^2 - 2*x^3 + 3*x^4 - 14*x^5 + 92*x^6 - 528*x^7 + 3027*x^8 - 18674*x^9 + 120414*x^10 - 790332*x^11 + 5264400*x^12 + ...
A(x)^2 / A(x^2 - 2*x*A(x)^2) = 1 + 6*x^2 - 24*x^3 + 120*x^4 - 696*x^5 + 4362*x^6 - 27720*x^7 + 179496*x^8 - 1188324*x^9 + 8004222*x^10 - 54609300*x^11 + 376571358*x^12 + ...
which also equals A(x)^3 / A(x^3 - 3*x*A(x)^3).
SPECIFIC VALUES.
A(t) = 1/9 at t = 0.121516943263807312205895948801335726496880098390997...
A(t) = 1/10 at t = 0.10853522544585482043493483988448061537900985360239...
  where 1/10 = A(t^3 - 3*t/10^3) / A(t^2 - 2*t/10^2).
A(1/8) = 0.11406786932603073004140288621992624859772243547241...
  where A(1/8) = A(1/8^3 - 3/8*A(1/8)^3) / A(1/8^2 - 2/8*A(1/8)^2).
A(1/9) = 0.10221632839303036250437950418981106643715860663579...
A(1/10) = 0.09261382926552257152263444179118404183078004746053...
A(1/11) = 0.08467305103076088485212166668503171895257965884840...
A(1/12) = 0.07799525781645435314575205510854905593664831977089...
A(-1/8) = -0.1721552141574965794714379396217931153561352397408...
A(-1/9) = -0.1381403570271418152616530595087372513703838778026...
A(-1/10) = -0.1188984739221037989978618004289786780166116550840...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..731

Crossrefs

Cf. A038464, A376226, A384270.

Programs

PARI

{a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^3 - 3*x*Ax^3 ) - Ax*subst(Ax, x, x^2 - 2*x*Ax^2 ), #A+1)); A[n+1]}
for(n=1, 25, print1(a(n), ", "))

Formula

a(n) ~ (-1)^(n+1) * c * d^n / n^(3/2), where d = 7.79529565596481925683030276174290122793451926009119... and c = 0.0350717712305315691918705178165312896756485548321... - Vaclav Kotesovec, Jul 16 2025