A212260 G.f. A(x) satisfies A(x)+A(x)^2+A(x)^3 = (1-sqrt(1-4*x))/2.

0, 1, 0, 1, 3, 5, 22, 64, 198, 710, 2332, 8105, 28665, 100653, 361104, 1301180, 4713267, 17217021, 63140534, 232702261, 861507251, 3200666821, 11933894310, 44636509320, 167427781950, 629691033738, 2373987233286, 8970240131032, 33965443165016, 128857452216256

Offset: 0

Author

Vladimir Kruchinin, May 12 2012

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A103779.

Programs

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 *)

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;

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