cp's OEIS Frontend

A116410 Expansion of (1-x-2x^2+sqrt(1-2x-3x^2))/(2*(1-2x-3x^2)).

%I A116410 #16 Jan 30 2020 21:29:15
%S A116410 1,1,3,8,23,66,192,561,1647,4850,14318,42351,125468,372191,1105275,
%T A116410 3285288,9772767,29090826,86646486,258208671,769820418,2296067565,
%U A116410 6850744365,20447143866,61045757604,182303186391,544550917797
%N A116410 Expansion of (1-x-2x^2+sqrt(1-2x-3x^2))/(2*(1-2x-3x^2)).
%C A116410 Partial sums are A116409.
%F A116410 a(n) = ((3^n+2*0^n)/3 + Sum_{k=0..floor(n/2)} C(n,2k)C(2k,k))/2.
%F A116410 From _Benedict W. J. Irwin_, May 30 2016: (Start)
%F A116410 Let y1(-1)=0, y1(0)=0, y1(1)=1,
%F A116410 Let (3n-3)*y1(n)+(2-5n)*y1(n+1)+(n-1)*y1(n+2)+(n+2)*y1(n+3)=0,
%F A116410 Let y2(-1)=0, y2(0)=0, y2(1)=1,
%F A116410 Let (n-1)*y2(n)+(2+n)*y2(n+1)-(5n+7)*y2(n+2)+(3n+6)*y2(n+3)=0,
%F A116410 a(n) = (4*3^n+3*(-1)^n*y1(n+1)+3^(n+2)*y2(n+1))/24, n>0.
%F A116410 (End)
%F A116410 a(n) ~ 3^(n-1)/2 * (1 + 3*sqrt(3/(Pi*n))/2). - _Vaclav Kotesovec_, May 30 2016
%F A116410 D-finite with recurrence: a(n) = ((5*n-4)*a(n-1) -(3*n-6)*a(n-2) -(9*n-18)*a(n-3))/n for n>3. - _Alois P. Heinz_, May 30 2016
%p A116410 a:= proc(n) option remember; `if`(n<4, [1$2, 3, 8][n+1],
%p A116410      ((5*n-4)*a(n-1)-(3*n-6)*a(n-2)-(9*n-18)*a(n-3))/n)
%p A116410     end:
%p A116410 seq(a(n), n=0..30);  # _Alois P. Heinz_, May 30 2016
%t A116410 Table[ (4 3^k + 3 (-1)^k DifferenceRoot[Function[{y, n}, {(-3 + 3 n) y[n] + (2 - 5 n) y[1 + n] + (-1 + n) y[2 + n] + (2 + n) y[3 + n] == 0, y[-1] == 0, y[0] == 0, y[1] == 1}]][1 + k] + 3^(2 + k)DifferenceRoot[Function[{y, n}, {(-1 + n) y[n] + (2 + n) y[1 + n] + (-7 - 5 n) y[2 + n] + (6 + 3 n) y[3 + n] == 0, y[-1] == 0, y[0] == 0, y[1] == 1}]][1 + k])/24, {k, 1, 20}] (* _Benedict W. J. Irwin_, May 30 2016 *)
%K A116410 easy,nonn
%O A116410 0,3
%A A116410 _Paul Barry_, Feb 13 2006