%I A024199
%S A024199 0,1,2,13,76,789,7734,110937,1528920,28018665,497895210,11110528485,241792844580,
%T A024199 6361055257725,163842638377950,4964894559637425,147721447995130800,5066706567801827025,
%U A024199 171002070002301095250,6548719685561840296125,247199273204273879989500
%N A024199 a(0) = 0, a(1) = 1, a(n+1) = 2*a(n) + (2*n-1)^2*a(n-1).
%C A024199 (2*n+1)!!/a(n+1), n>=0, is the n-th approximant for William Brouncker's 
               continued fraction for 4/Pi =1+1^2/(2+3^2/(2+5^2/(2+... See the C. 
               Brezinski and J.-P. Delahaye references given under A142969 and A142970, 
               respectively. The double factorials (2*n+1)!! = A001147(n+1) enter. 
               W. Lang, Oct 06 2008. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Oct 06 2008]
%D A024199 A. E. Jolliffe, Continued Fractions, in Encyclopaedia Britannica, 11th 
               ed., pp. 30-33; see p. 31.
%F A024199 a(n) = s(1)s(2)...s(n)(1/s(1) - 1/s(2) + ... + c/s(n)) where c=(-1)^(n+1) 
               and s(k) = 2k-1 for k = 1, 2, 3, ...
%F A024199 A024199(n) + A024200(n) = A001147(n) = (2n-1)!! - Max Alekseyev (maxale(AT)gmail.com), 
               Sep 23 2007.
%F A024199 A024199(n)/A024200(n) -> Pi/(4-Pi) as n -> oo. - Max Alekseyev (maxale(AT)gmail.com), 
               Sep 23 2007.
%F A024199 Contribution from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Oct 06 2008: (Start)
%F A024199 E.g.f. for a(n+1), n>=0: (sqrt(1-2*x)+arcsin(2*x)*sqrt(1+2*x)/2)/((1-4*x^2)^(1/
               2)*(1-2*x)). From the recurrence, solving (1-4*x^2)y''(x)-2*(8*x+1)*y'(x)-9*y=0 
               with inputs y(0)=1, y'(0)=2.
%F A024199 a(n+1)= A003148(n) + A143165(n), n>=0 (from the two terms of the e.g.f.). 
               (End)
%F A024199 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 
               2009: (Start)
%F A024199 a(n) = (-1)^(n-1)*(2*n-3)!! + (2*n-1)*a(n-1) with a(0) = 0.
%F A024199 a(n) = (2*n-1)!!*sum((-1)^(k)/(2*k+1), k=0..n-1)
%F A024199 (End)
%p A024199 f := proc(n) option remember; local a,b,t1,t2,t3,i,j,k; a := 0; b := 
               1; if n=0 then RETURN(a) elif n=1 then RETURN(b) else RETURN(2*f(n-1)+ 
               (2*n-3)^2*f(n-2)); fi; end;
%Y A024199 Cf. A004041.
%Y A024199 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 
               2009: (Start)
%Y A024199 Cf. A007509 and A025547.
%Y A024199 Equals first column of A167584.
%Y A024199 Equals row sums of A167591.
%Y A024199 Equals first right hand column of A167594.
%Y A024199 (End)
%Y A024199 Sequence in context: A161130 A007509 A077413 this_sequence A037523 A037732 
               A090187
%Y A024199 Adjacent sequences: A024196 A024197 A024198 this_sequence A024200 A024201 
               A024202
%K A024199 nonn,new
%O A024199 0,3
%A A024199 Clark Kimberling (ck6(AT)evansville.edu)
%E A024199 Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 19 2002.