A137265 - OEIS

%I #24 Nov 02 2023 10:39:46

%S 1,1,2,8,35,163,796,4024,20885,110654,596064,3254752,17974893,

%T 100227022,563482140,3190633232,18179765509,104158703503,599698459613,

%U 3467978715612,20134256546896,117313279477959,685756774642494,4020515276730588,23636036336651811

%N G.f. y(x) is solution of x y^3 - (1 + x^2) y + 1 = 0 with y(0) = 1.

%H Vincenzo Librandi, <a href="/A137265/b137265.txt">Table of n, a(n) for n = 0..100</a>

%F a(0) = 1, a(1) = 1, a(n) = -a(n-2) + sum_{i=0}^{n-1} sum_{j=0}^{n-1-i} a(i) a(j) a(n-1-i-j).

%F a(n) ~ sqrt(1 - (2*r)^(5/3)) / (2^(4/3) * sqrt(3*Pi) * n^(3/2) * r^(n + 1/3)), where r = 0.15978798947663136723274504893788499231133813071845... is the real root of the equation (1+r^2)^3 = 27*r/4. - _Vaclav Kotesovec_, May 03 2016

%F a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k) / (2*n-4*k+1). - _Seiichi Manyama_, Nov 02 2023

%e a(3) = 8 because g(x) = 1 + x + 2 x^2 + 8 x^3 + O(x^4) satisfies x*g(x)^3 - (1 + x^2)*g(x) + 1 = O(x^4).

%p f:= (x,y) -> x*y^3 - (1 + x^2)*y + 1; N:= (y,n) -> convert(normal(taylor(y-f(x,y)/D[2](f)(x,y),x=0,n)),polynom); Y:= 1; for j from 1 to 6 do Y:= N(Y,2^j) end do; seq(coeftayl(Y,x=0,j),j=0..2^6-1);

%t max = 22; g[x_] := Sum[a[k]*x^k, {k, 0, max}]; coes = CoefficientList[ Series[ x*g[x]^3 - (1+x^2)*g[x] + 1, {x, 0, max}], x]; sol = First[ Solve[ Thread[ coes == 0 ] ] ]; Table[a[n] /. sol, {n, 0, max}](* _Jean-François Alcover_, Nov 28 2011 *)

%t terms = 25; y[_] = 1; Do[y[x_] = (1 + x*y[x]^3)/(1 + x^2) + O[x]^terms, terms]; CoefficientList[y[x], x] (* _Jean-François Alcover_, Jan 11 2018 *)

%o (PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(3*n-5*k, k)*binomial(3*n-6*k, n-2*k)/(2*n-4*k+1)); \\ _Seiichi Manyama_, Nov 02 2023

%Y Cf. A000108, A200753, A200755.

%Y Cf. A364474.

%K nonn,easy

%O 0,3

%A _Robert Israel_, Mar 12 2008