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G.f.: A(x) satisfies: A( 5*x - 4*A(x) ) = x - 9*x^2.
8

%I #12 Oct 09 2017 09:07:27

%S 1,3,-24,624,-23040,1102848,-63320064,4204118016,-315301920768,

%T 26294301622272,-2410825799172096,240940701368451072,

%U -26069519382431662080,3036630253199488450560,-378975438359301231476736,50464427995254851962404864,-7143619061783407156694876160,1071471673355571318056420376576,-169778310385940944720470768353280,28343313796900390128748827210940416,-4972974647067348293267827631390195712

%N G.f.: A(x) satisfies: A( 5*x - 4*A(x) ) = x - 9*x^2.

%H Paul D. Hanna, <a href="/A292813/b292813.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) ~ (-1)^n * c * 6^n * n! / (n^(1/2 + 2*log(2)/3) * (log(2))^n), where c = 0.04600302526929... - _Vaclav Kotesovec_, Oct 09 2017

%e G.f.: A(x) = x + 3*x^2 - 24*x^3 + 624*x^4 - 23040*x^5 + 1102848*x^6 - 63320064*x^7 + 4204118016*x^8 - 315301920768*x^9 + 26294301622272*x^10 - 2410825799172096*x^11 + 240940701368451072*x^12 - 26069519382431662080*x^13 + 3036630253199488450560*x^14 - 378975438359301231476736*x^15 +...

%e such that A( 5*x - 4*A(x) ) = x - 9*x^2.

%e RELATED SERIES.

%e Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins:

%e Ai(x) = x - 3*x^2 + 42*x^3 - 1119*x^4 + 41670*x^5 - 1961790*x^6 + 110327892*x^7 - 7171123959*x^8 + 527249131182*x^9 - 43195371425274*x^10 +...

%e where Ai(x - 9*x^2) = 5*x - 4*A(x).

%o (PARI) {a(n) = my(A=x,V=[1,3]); for(i=1,n, V = concat(V,0); A=x*Ser(V); V[#V] = Vec( subst(A,x, 5*x - 4*A) )[#V]/3 );V[n]}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A291198, A292812, A292814, A292811, A293454, A293455, A293456.

%K sign

%O 1,2

%A _Paul D. Hanna_, Sep 23 2017