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A273162 G.f. A(x) satisfies: A(x*B(x)) = x^2 - x^3, where A(B(x)) = x. 3

%I

%S 1,1,3,8,28,95,351,1309,5056,19787,78847,317705,1294673,5321598,

%T 22047985,91957296,385832452,1627351862,6896087775,29345806842,

%U 125353612440,537303633158,2310270577888,9962069922553,43070357677938,186663591654655,810799482641934,3529141491880136,15390864728209348,67241645122541334,294268407263148650,1289830017807048396,5661906416526228762,24888336664147079838

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

%H Paul D. Hanna, <a href="/A273162/b273162.txt">Table of n, a(n) for n = 1..515</a>

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

%e G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 28*x^5 + 95*x^6 + 351*x^7 + 1309*x^8 + 5056*x^9 + 19787*x^10 + 78847*x^11 + 317705*x^12 +...

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

%e RELATED SERIES.

%e A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 22*x^5 + 81*x^6 + 294*x^7 + 1124*x^8 + 4338*x^9 + 17140*x^10 + 68476*x^11 + 277229*x^12 + 1132716*x^13 + 4669477*x^14 +...

%e A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 43*x^6 + 168*x^7 + 648*x^8 + 2574*x^9 + 10284*x^10 + 41691*x^11 + 170333*x^12 + 702273*x^13 + 2915100*x^14 +...

%e A(x*A(x)) = A(x)^2 - A(x)^3 where

%e A(x)^2 - A(x)^3 = x^2 + x^3 + 4*x^4 + 10*x^5 + 38*x^6 + 126*x^7 + 476*x^8 + 1764*x^9 + 6856*x^10 + 26785*x^11 + 106896*x^12 + 430443*x^13 + 1754377*x^14 +...

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

%e B(x) = x - x^2 - x^3 + 2*x^4 - 2*x^5 + 3*x^6 - x^7 - 7*x^8 + 10*x^9 + 2*x^10 - 15*x^11 + 2*x^12 + 34*x^13 - 51*x^14 + 17*x^15 + 73*x^16 - 218*x^17 + 323*x^18 - 135*x^19 - 467*x^20 + 1139*x^21 - 1279*x^22 + 430*x^23 + 1587*x^24 +...

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

%e also, B(x) = B(x^2 - x^3)/x.

%o (PARI) {a(n) = my(A=[1, 1], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(F, x, x*B))[#A]); A[n]}

%o for(n=1, 50, print1(a(n), ", "))

%Y Cf. A273203, A273095.

%K nonn

%O 1,3

%A _Paul D. Hanna_, May 16 2016

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Last modified August 9 13:57 EDT 2020. Contains 336323 sequences. (Running on oeis4.)