OFFSET
0,5
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = (sum(k=2..n, C(2*n-k-1,n-1)*(sum(i=1..k-1, (-1)^i*C(i,k-i-1) * C(k+i-1,k-1)))) +C(2*n-2,n-1))/n if n>0, a(0) = 0.
a(n) ~ c * 4^n / (sqrt(Pi) * n^(3/2)), where c = 0.12273243737616788383659461976... is the real root of the equation 8*c*(134*c^2 - 1) = 1. - Vaclav Kotesovec, Nov 20 2017
MAPLE
a:= n-> coeff(series(RootOf(A+A^2+A^3=(1-sqrt(1-4*x))/2, A), x, n+1), x, n): seq(a(n), n=0..40); # Alois P. Heinz, May 12 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [0, 1, 0, 1, 3][n+1],
(2*(n-1)*(n-2)*(n-3)*(21574*n^2-148237*n+252420)*a(n-1)
+(n-2)*(n-3)*(30485*n^3-173514*n^2+191353*n+116820)*a(n-2)
-4*(n-3)*(12730*n^4-266121*n^3+1766621*n^2-4771248*n+4563630)*a(n-3)
-36*(6*n-23)*(67*n-183)*(6*n-25)*(3*n-10)*(3*n-11)*a(n-4))/
(132*(67*n-250)*(n-1)*(n-2)*(n-3)*n))
end:
seq(a(n), n=0..40); # Alois P. Heinz, Sep 27 2013
MATHEMATICA
nn=29; f[x_]:=Sum[a[n]x^n, {n, 0, nn}]; sol=SolveAlways[0==Series[f[x]+f[x]^2 +f[x]^3-(1-(1-4x)^(1/2))/2, {x, 0, nn}], x][[3]]; Table[a[n], {n, 0, nn}]/.sol (* Geoffrey Critzer, Sep 27 2013 *)
PROG
(Maxima) a(n):=(sum(binomial(2*n-k-1, n-1)*(sum((-1)^i*binomial(i, k-i-1) *binomial(k+i-1, k-1), i, 1, k-1)), k, 2, n) +binomial(2*n-2, n-1))/n;
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, May 12 2012
STATUS
approved