A137265 G.f. y(x) is solution of x y^3 - (1 + x^2) y + 1 = 0 with y(0) = 1.

1, 1, 2, 8, 35, 163, 796, 4024, 20885, 110654, 596064, 3254752, 17974893, 100227022, 563482140, 3190633232, 18179765509, 104158703503, 599698459613, 3467978715612, 20134256546896, 117313279477959, 685756774642494, 4020515276730588, 23636036336651811

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Formula

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).

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

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

Example

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).

Maple

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);

Mathematica

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 *)
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 *)

Prog

(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

Crossrefs

Cf. A000108, A200753, A200755.

Cf. A364474.

Sequence in context: A082759 A243204 A279013 * A364472 A303070 A129580

Adjacent sequences: A137262 A137263 A137264 * A137266 A137267 A137268

Keyword

nonn,easy

Author

Robert Israel, Mar 12 2008

Status

approved