A276364 - OEIS

%I #13 Nov 06 2019 12:40:43

%S 1,2,18,252,4410,88734,1969668,47104056,1195658550,31891944750,

%T 887565934494,25639389304560,765765781572600,23574635888791804,

%U 746297727831434376,24247096863466015152,807243935471150901066,27503153109167182217082,957899411829034037383374,34073454839478198669105444,1236879534288183156526996062,45788365378826408823663436974,1727576456033196960394178300184

%N G.f. A(x) satisfies: A(x - A(x)^3) = x + A(x)^3, where A(x) = Sum_{n>=1} a(n)*x^(2*n-1).

%H Paul D. Hanna, <a href="/A276364/b276364.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) also satisfies:

%F (1) A(x) = x + 2 * A( x/2 + A(x)/2 )^3.

%F (2) A(x) = -x + 2 * Series_Reversion(x - A(x)^3).

%F (3) R(x) = -x + 2 * Series_Reversion(x + A(x)^3), where R(A(x)) = x.

%F (4) R( ( x/2 - R(x)/2 )^(1/3) ) = x/2 + R(x)/2, where R(A(x)) = x.

%e G.f.: A(x) = x + 2*x^3 + 18*x^5 + 252*x^7 + 4410*x^9 + 88734*x^11 + 1969668*x^13 + 47104056*x^15 + 1195658550*x^17 + 31891944750*x^19 + 887565934494*x^21 + 25639389304560*x^23 + 765765781572600*x^25 +...

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

%e RELATED SERIES.

%e Note that Series_Reversion(x - A(x)^3) = x/2 + A(x)/2, which begins:

%e Series_Reversion(x - A(x)^2) = x + x^3 + 9*x^5 + 126*x^7 + 2205*x^9 + 44367*x^11 + 984834*x^13 + 23552028*x^15 + 597829275*x^17 + 15945972375*x^19 +...

%e Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,

%e R(x) = x - 2*x^3 - 6*x^5 - 60*x^7 - 830*x^9 - 13950*x^11 - 267156*x^13 - 5629752*x^15 - 127807290*x^17 - 3082830030*x^19 - 78254901810*x^21 - 2076067799280*x^23 - 57266880966792*x^25 +...

%e then Series_Reversion(x + A(x)^3) = x/2 + R(x)/2.

%e Also, A(x) = x + 2 * A( x/2 + A(x)/2 )^3, where

%e A( x/2 + A(x)/2 ) = x + 3*x^3 + 33*x^5 + 528*x^7 + 10235*x^9 + 224001*x^11 + 5343738*x^13 + 136167888*x^15 + 3659113701*x^17 + 102800460825*x^19 + 3001057504233*x^21 + 90627712970220*x^23 + 2821487673544920*x^25 +...

%e and

%e A( x/2 + A(x)/2 )^3 = x^3 + 9*x^5 + 126*x^7 + 2205*x^9 + 44367*x^11 + 984834*x^13 + 23552028*x^15 + 597829275*x^17 + 15945972375*x^19 +...

%e which equals -x/2 + A(x)/2.

%t nmin = 1; nmax = 60; sol = {b[1] -> 1}; nsol = Length[sol];

%t Do[A[x_] = Sum[b[k] x^k, {k, 0, n}] /. sol;eq = CoefficientList[A[x - A[x]^3] - x - A[x]^3 + O[x]^(n+1), x][[nsol+1;;]] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, nsol + 1, nmax}];

%t a[n_] := b[2n-1];

%t a /@ Range[nmin, (nmax+1)/2 // Floor] /. sol (* _Jean-François Alcover_, Nov 06 2019 *)

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

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

%Y Cf. A153851, A275765.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 31 2016