A243019 Expansion of -(2*x*sqrt(1-8*x^2)-2*x) / (16*x^3+sqrt(1-8*x^2)*(4*x^2+2*x-1)-8*x^2-2*x+1).

1, 1, 5, 7, 33, 51, 233, 379, 1697, 2851, 12585, 21627, 94449, 165075, 714873, 1266027, 5445441, 9746883, 41687369, 75275227, 320420753, 582881971, 2471008281, 4523575371, 19108837601, 35174066851

Offset: 0

Author

Vladimir Kruchinin, May 29 2014

Keywords

• nonn
• changed

Comments

Number of ternary strings of length n that have the same number or more 0's than the combined number of 1's and 2's. For example, a(4) = 33 since the strings are (number of permutations in parentheses): 0000 (1), 0001 (4), 0002 (4), 0011 (6), 0022 (6), 0012 (12). - Enrique Navarrete, Aug 14 2025

Crossrefs

Cf. A000108, A013609, A128418.

Programs

Mathematica

CoefficientList[Series[-(2*x*Sqrt[1-8*x^2]-2*x)/(16*x^3+Sqrt[1-8*x^2]*(4*x^2+2*x-1)-8*x^2-2*x+1),{x,0,20}],x] (* Vaclav Kotesovec, May 29 2014 *)

Maxima

a(n):=sum(2^(i)*binomial(n,i),i,0,floor((n)/2));

Formula

a(n) = sum(i=0..floor(n/2), 2^i*binomial(n,i)).

G.f.: (x*C'(2*x^2))/(C(2*x^2)*(1-x*C(2*x^2))), where C(x) is g.f. of A000108.

a(n) ~ 2^(3*n/2) * (2+sqrt(2) + (-1)^n*(2-sqrt(2))) / sqrt(2*Pi*n). - Vaclav Kotesovec, May 29 2014

D-finite with recurrence: n^2*a(n) = (3*n^2-4)*a(n-1) + 4*(2*n^2 - 2*n - 1)*a(n-2) - 24*(n-2)*(n+1)*a(n-3). - Vaclav Kotesovec, May 29 2014

a(n) = Sum_{k=0..floor(n/2)} A013609(n,k). - Enrique Navarrete, Aug 14 2025