|
%I #8 Oct 10 2017 07:31:29
%S 1,1,4,21,136,941,6862,51998,405256,3228208,26167124,215135798,
%T 1789732224,15037735467,127429439672,1087804527493,9345903237508,
%U 80751170130654,701224835836174,6116663059682404,53570252214591940,470886419343198751,4152853930912625384,36735754728365394280,325861321839388361170,2897885772356354420332
%N G.f. A(x) satisfies: A( x^2*A(x) - A(x)^4 ) = x^3.
%H Paul D. Hanna, <a href="/A292928/b292928.txt">Table of n, a(n) for n = 1..520</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 9.4477112726509666047269249019968778065... and c = 0.01614622962039842631454... - _Vaclav Kotesovec_, Oct 10 2017
%e G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 136*x^5 + 941*x^6 + 6862*x^7 + 51998*x^8 + 405256*x^9 + 3228208*x^10 + 26167124*x^11 + 215135798*x^12 + 1789732224*x^13 + 15037735467*x^14 + 127429439672*x^15 + 1087804527493*x^16 +...
%e such that A( x^2*A(x) - A(x)^4 ) = x^3.
%e RELATED SERIES.
%e x^2*A(x) - A(x)^4 = x^3 - x^6 - 2*x^9 - 6*x^12 - 32*x^15 - 143*x^18 - 718*x^21 - 3866*x^24 - 21054*x^27 - 118409*x^30 +...
%e Let B(x) be the series reversion of A(x), so that B(A(x)) = x, then
%e B(x) = x - x^2 - 2*x^3 - 6*x^4 - 32*x^5 - 143*x^6 - 718*x^7 - 3866*x^8 - 21054*x^9 - 118409*x^10 - 679560*x^11 - 3955972*x^12 - 23343650*x^13 - 139235773*x^14 - 838057508*x^15 - 5084422207*x^16 +...
%e then x^2*A(x) - A(x)^4 = B(x^3).
%o (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -Vec(subst(F, x, x^2*F - F^4))[#A] ); polcoeff(A, n)}
%o for(n=1, 40, print1(a(n), ", "))
%Y Cf. A265940, A268039, A272463, A292927.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Sep 26 2017
|
