A273955 - OEIS

%I #15 Jun 24 2016 11:59:14

%S 1,1,-1,2,-4,12,-36,115,-366,1202,-4016,13684,-47192,164550,-578773,

%T 2051994,-7324990,26306860,-94980720,344555898,-1255235744,4590432218,

%U -16845658552,62014596756,-228956736888,847546307549,-3145089430938,11697191534690,-43595085475847,162793363083734,-609005952973882,2282129119421879,-8565364253229324,32195552437196082,-121185775973925826,456749227410641398

%N G.f. A(x) satisfies: A( x*A(x) - A(x)^2 ) = -x^3.

%H Paul D. Hanna, <a href="/A273955/b273955.txt">Table of n, a(n) for n = 1..300</a>

%F If A(B(x)) = x, then g.f. A(x) and B(x) satisfy:

%F (1) x*A(x) - A(x)^2 = B(-x^3).

%F (2) A(x) = x - x*C( B(-x^3)/x^2 ), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

%F a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 3.9374997379511376037..., c = 0.034997955229443779... . - _Vaclav Kotesovec_, Jun 24 2016

%e G.f.: A(x) = x + x^2 - x^3 + 2*x^4 - 4*x^5 + 12*x^6 - 36*x^7 + 115*x^8 - 366*x^9 + 1202*x^10 - 4016*x^11 + 13684*x^12 - 47192*x^13 + 164550*x^14 +...

%e such that A( x*A(x) - A(x)^2 ) = -x^3.

%e RELATED SERIES.

%e A(x)^2 = x^2 + 2*x^3 - x^4 + 2*x^5 - 3*x^6 + 12*x^7 - 36*x^8 + 118*x^9 - 366*x^10 + 1202*x^11 - 4004*x^12 + 13684*x^13 - 47192*x^14 + 164604*x^15 +...

%e x*A(x) - A(x)^2 = -x^3 - x^6 - 3*x^9 - 12*x^12 - 54*x^15 - 264*x^18 - 1362*x^21 - 7300*x^24 - 40245*x^27 - 226746*x^30 - 1299779*x^33 +...

%e Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then

%e B(x) = x - x^2 + 3*x^3 - 12*x^4 + 54*x^5 - 264*x^6 + 1362*x^7 - 7300*x^8 + 40245*x^9 - 226746*x^10 + 1299779*x^11 - 7556310*x^12 + 44445150*x^13 - 264010326*x^14 + 1581537357*x^15 - 9543458802*x^16 + 57956158488*x^17 - 353941849554*x^18 +...

%e such that x*A(x) - A(x)^2 = B(-x^3).

%o (PARI) {a(n) = my(A=[1, 1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = polcoeff(x^3 + subst(F, x, x*F - F^2), #A+1) ); A[n]}

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A272411.

%K sign

%O 1,4

%A _Paul D. Hanna_, Jun 12 2016