A025231 - OEIS

%I #39 Apr 21 2023 10:43:28

%S 2,3,12,57,300,1686,9912,60213,374988,2381322,15361896,100389306,

%T 663180024,4421490924,29712558576,201046204173,1368578002188,

%U 9366084668802,64403308499592,444739795023054,3082969991029800

%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3.

%H Vincenzo Librandi, <a href="/A025231/b025231.txt">Table of n, a(n) for n = 1..200</a>

%F G.f.: (1 - sqrt(1 - 8*x + 4*x^2))/2. - _Michael Somos_, Jun 08 2000

%F n*a(n) = (8*n - 12)*a(n - 1) - (4*n - 12)*a(n - 2). [_Richard Choulet_, Dec 16 2009]

%F G.f.: 1 + x - G(0); G(k) = 1 + 2*x - 3*x/G(k + 1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Jan 05 2012

%F a(n) ~ sqrt(4*sqrt(3) - 6)*(4 + 2*sqrt(3))^n/(4*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2012

%F G.f.: x + x/W(0), where W(k)= 1 - 2*x - x/W(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Aug 16 2013 [Edited by _Michael Somos_, Apr 10 2014]

%F 0 = a(n)*(+16*a(n+1) - 80*a(n+2) + 16*a(n+3)) + a(n+1)*(+16*a(n+1) + 56*a(n+2) - 20*a(n+3)) + a(n+2)*(+4*a(n+2) + a(n+3)) if n>0. - _Michael Somos_, Apr 10 2014

%F a(n) = Sum_{k=0..n} C(k+1,n-k)*2^(2*k+1-n)*(-1)^(n-k)*C(2*k,k)/(k+1). - _Vladimir Kruchinin_, Apr 21 2023

%e G.f. = 2*x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1686*x^6 + 9912*x^7 + ...

%t Table[SeriesCoefficient[(1 - Sqrt[1 - 8*x + 4*x^2])/2,{x, 0, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Oct 07 2012 *)

%o (PARI) a(n)=polcoeff((1-sqrt(1-8*x+4*x^2+x*O(x^n)))/2,n)

%o (Maxima)

%o a(n):=sum(binomial(k+1,n-k)*2^(2*k+1-n)*(-1)^(n-k)*binomial(2*k,k)/(k+1),k,0,n); /* __Vladimir Kruchinin_, Apr 21 2023 */

%Y Essentially the same as A047891.

%K nonn,eigen

%O 1,1

%A _Clark Kimberling_