A025259 - OEIS

%I #18 Jan 16 2025 11:15:56

%S 2,-1,1,2,3,5,11,27,65,154,371,917,2303,5828,14839,38049,98237,255106,

%T 665655,1744318,4588881,12115727,32092597,85259160,227117149,

%U 606512641,1623412007,4354539253,11703442343,31512560472,84996156243,229621401075

%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 2, -1, 1, 2.

%H Vincenzo Librandi, <a href="/A025259/b025259.txt">Table of n, a(n) for n = 1..200</a>

%F G.f.: (1+2x-x^2-sqrt(1-4x+6x^2-8x^3+x^4))/(2x)=2-x+x^2*c(x^2/(1-x)^4)/(1-x)^2, c(x) the g.f. of A000108; a(n+3)=sum{k=0..floor(n/2), C(n+3k+1,6k+1)C(k)}, where C(n)=A000108(n). - _Paul Barry_, May 31 2006

%F Conjecture: n*a(n) +2*(3-2*n)*a(n-1) +6*(n-3)*a(n-2) +4*(9-2*n)*a(n-3) +(n-6)*a(n-4)=0. - _R. J. Mathar_, Sep 29 2012

%F a(n) ~ sqrt(3*r^2-2*r^3-3*r+1) / (2 * sqrt(Pi) * n^(3/2) * r^(n+1)), where r = 2 + 1/sqrt(2/(6 + (9 + sqrt(273))^(1/3) / 3^(2/3) - 4/(3*(9 + sqrt(273)))^(1/3))) - 1/2*sqrt(24 - (2*(9 + sqrt(273))^(1/3)) / 3^(2/3) + 8/(3*(9 + sqrt(273)))^(1/3) + 44*sqrt(2/(6 + (9 + sqrt(273))^(1/3) / 3^(2/3) - 4/(3*(9 + sqrt(273)))^(1/3)))) = 0.352722901375863... is the root of the equation 1-4*r+6*r^2-8*r^3+r^4=0. - _Vaclav Kotesovec_, Feb 01 2014

%t CoefficientList[Series[(1+2*x-x^2-Sqrt[1-4*x+6*x^2-8*x^3+x^4])/(2*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2014 *)

%K sign

%O 1,1

%A _Clark Kimberling_