A256169 Expansion of (1-sqrt(1-4*(x+x^2)^2))/(2*(x+x^2)^2).

1, 0, 1, 2, 3, 8, 17, 38, 91, 212, 509, 1234, 3007, 7408, 18353, 45742, 114643, 288620, 729749, 1852138, 4716951, 12050920, 30876185, 79317990, 204256027, 527171556, 1363428637, 3533070818, 9171798815, 23849951200, 62116162081

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=floor((n-1)/4)..(n-1)} binomial(2*k,n-2*k-1)*C(k)}, where C(k) are the Catalan numbers (A000108).

G.f. g(x) satisfies 1 - g(x) + x^2 (1+x)^2 g(x)^2 = 0. Recurrence: a(n) = sum(j>=0, a(j)*(a(n-j-2)+2*a(n-j-3)+a(n-j-4))) for n >= 1, where a(j) = 0 for j < 0. - Robert Israel, Mar 17 2015

D-finite with recurrence: (n+2)*a(n) = -(n+3)*a(n-1) + 4*(n-1)*a(n-2) + 4*(3*n - 5)*a(n-3) + 4*(3*n - 7)*a(n-4) + 4*(n-3)*a(n-5). - Vaclav Kotesovec, Mar 17 2015

a(n) ~ sqrt(6-2*sqrt(3)) * (1+sqrt(3))^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2015

Maple

f:= proc(n) option remember;
    add(procname(j)*(procname(n-j-2)+2*procname(n-j-3)+procname(n-j-4)), j=0..n-2)
end proc:
f(0):= 1: f(-1):= 0: f(-2):= 0:
seq(f(n), n=0..100); # Robert Israel, Mar 17 2015

Mathematica

CoefficientList[Series[(1-Sqrt[1-4*(x+x^2)^2])/(2*(x+x^2)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 17 2015 *)

Prog

(Maxima)
a(n):=sum((binomial(2*k, n-2*k-1)*binomial(2*k, k))/(k+1), k, floor((n-1)/4), n-1);
(PARI) default(seriesprecision, 50); Vec((1-sqrt(1-4*(x+x^2)^2))/(2*(x+x^2)^2) + O(x^50)); \\ Michel Marcus, Mar 17 2015

Crossrefs

Cf. A000108.

Sequence in context: A292401 A132333 A182889 * A298405 A219788 A099965

Adjacent sequences: A256166 A256167 A256168 * A256170 A256171 A256172

Keyword

nonn

Author

Vladimir Kruchinin, Mar 17 2015

Status

approved