OFFSET
0,3
COMMENTS
Define e.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!. Then A(x) = P(z + x/sqrt(6)) where P is the Weierstrass P-function with g_2 = 0, g_3 = 4 and z = 2.10327313773966...*i satisfies P(z) = 1, P'(z) = 0. - Michael Somos, Apr 18 2022
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..270
Markus Kuba, Alois Panholzer, Combinatorial families of multilabelled increasing trees and hook-length formulas, arXiv:1411.4587 [math.CO], 2014.
FORMULA
E.g.f. satisfies: A(x) = exp( Integral 1/A(x) * Integral A(x)^2 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(1*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015
From Vaclav Kotesovec, Jun 14 2015: (Start)
a(n) ~ c * d^n * n!^2 * sqrt(n), where d = A258895 = 32*Pi / (Gamma(1/6) * Gamma(1/3))^2 = 2^(17/3) * Pi^2 / (3*Gamma(1/3)^6) = 0.452104299183420528841..., c = 1.53043521765866544548745... = 2^(20/3) * Pi^(3/2) / Gamma(1/3)^6 = 192*sqrt(Pi) / (Gamma(1/3)*Gamma(1/6))^2.
a(n) ~ 3 * 2^(5*n+7) * Pi^(n+3/2) * n^(2*n+3/2) / (exp(2*n) * Gamma(1/6)^(2*n+2) * Gamma(1/3)^(2*n+2)).
a(n) ~ 2^((17*n+23)/3) * Pi^(2*n+5/2) * n^(2*n+3/2) / (3^n * exp(2*n) * Gamma(1/3)^(6*n+6)).
(End)
Define e.g.f. A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!. Then 3*A'(x)^2 = 2*A(x)^3 - 2. - Michael Somos, Jan 07 2022
EXAMPLE
a(3) = 1*a(0)a(2) + 6*a(1)a(1) + 1*a(2)a(0) = 2+6+2 = 10.
E.g.f.: A(x) = 1 + x^2/2! + 2*x^4/4! + 10*x^6/6! + 80*x^8/8! + ...
G.f. = 1 + x + 2*x^2 + 10*x^3 + 80*x^4 + 1000*x^8 + 17600*x^9 + ...
MATHEMATICA
Clear[a]; a[n_]:=a[n]=If[n<2, 1, Sum[a[j]*a[n-1-j]*Binomial[2*n-2, 2*j], {j, 0, n-1}]]; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Jun 14 2015 *)
a[ n_] := If[n < 0, 0, Module[{x, y}, Nest[Dt[#] /. {Dt[x] -> y, Dt[y] -> x^2} &, x, 2*n] /. {x -> 1, y -> 0}]]; (* Michael Somos, Apr 18 2022 *)
PROG
(PARI) /* E.g.f. A(x) = exp( Integral 1/A(x) * Integral A(x)^2 dx dx ) */
{a(n) = local(A=1+x); for(i=1, n, A = exp( intformal( 1/A * intformal( A^2 + x*O(x^n)) ) ) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(2*n), ", ")) \\ Paul D. Hanna, Jun 02 2015
(PARI) /* By definition: */
{a(n) = if(n==0, 1, sum(k=0, n-1, binomial(2*n-2, 2*k)*a(k)*a(n-k-1)))}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jun 02 2015
(PARI) {a(n) = if(n<0, 0, my(x='x, y = 1 + O(x^2)); for(i=1, n, y = 1 + intformal(intformal(y^2))); polcoeff(y, 2*n)*(2*n)!)}; /* Michael Somos, Apr 18 2022 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Henry Bottomley, Aug 30 2001
EXTENSIONS
More terms from Vaclav Kotesovec, Jun 14 2015
STATUS
approved