A259206 G.f. satisfies: A(x) = (1-x^2) * (1 + x*A(x)^2).

1, 1, 1, 2, 4, 10, 26, 71, 198, 562, 1616, 4698, 13786, 40784, 121516, 364341, 1098522, 3328713, 10131858, 30963944, 94975810, 292291370, 902277354, 2793028146, 8668118364, 26965143756, 84068366948, 262631106850, 822019561552, 2577421479004

Offset

0,4

Links

Table of n, a(n) for n=0..29.

Formula

G.f.: (1 - sqrt(1 - 4*x*(1-x^2)^2)) / (2*x*(1-x^2)).

G.f.: (1-x^2) * C( x*(1-x^2)^2 ), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers.

Conjecture: (n+1)*a(n) +(n+2)*a(n-1) +(-21*n+29)*a(n-2) +(7*n-30)*a(n-3) +30*(2*n-7)*a(n-4) +6*(-2*n+9)*a(n-5) +30*(-2*n+11)*a(n-6) +2*(2*n-13)*a(n-7) +10*(2*n-15)*a(n-8)=0. - R. J. Mathar, Jun 07 2016

Recurrence: (n+1)*a(n) = 2*(2*n - 1)*a(n-1) + (n+1)*a(n-2) - 6*(2*n - 5)*a(n-3) + 6*(2*n - 9)*a(n-5) - 2*(2*n - 13)*a(n-7). - Vaclav Kotesovec, Nov 20 2017

a(n) ~ sqrt((s^2*(1 - 3*r^2) - 2*r) / (Pi*(1 - r^2))) / (2*n^(3/2)*r^n), where r = 0.3031676052060370647559465864486937555025708097883... and s = 1.816178806307272896061936511407203981069565469811... are roots of the system of equations s + (-1 + r^2)*(1 + r*s^2) = 0, 1 + 2*r^3*s = 2*r*s. - Vaclav Kotesovec, Nov 20 2017

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 26*x^6 + 71*x^7 +...

Mathematica

CoefficientList[Series[(1 - Sqrt[1 - 4*x + 8*x^3 - 4*x^5]) / (2*x*(1 - x^2)), {x, 0, 30}], x] (* Vaclav Kotesovec, Nov 20 2017 *)

Prog

(PARI) {a(n) = local(A=1+x); for(i=1, 31, A = (1+x*A^2)*(1-x^2) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Sequence in context: A071962 A257520 A035079 * A106221 A149811 A149812

Adjacent sequences: A259203 A259204 A259205 * A259207 A259208 A259209

Keyword

nonn

Author

Paul D. Hanna, Jun 27 2015

Status

approved