A008563 - OEIS

%I #21 Mar 10 2020 18:50:07

%S 2,80,38400,77875200,449711308800,5935407169536000,

%T 155268186793574400000,7282309435397819596800000,

%U 568366274740400734548787200000,69683365216674756749626756300800000,12818556197716658801397242655670272000000,3408069131098061826431206876376696094720000000

%N Coefficients of series arising in solution of Riccati equation y' = y^2 + x^2.

%D G. N. Watson, Bessel Functions, p. 2.

%H H. G. Ellis, <a href="https://projecteuclid.org/euclid.rmjm/1250130951">Continued fraction solutions of the general Riccati differential equation</a>, Rocky Mountain Journal of Mathematics, Vol. 4, Number 2, 353-356, Spring 1974.

%F b(0)=b(1)=b(2)=0, b(3)=2, b(n+1) = Sum_{k=1..n} C(n,k)*b(k)*b(n-k), a(n)=b(4n-1).

%F a(1) = 2, a(n+1) = Sum_{k=1..n} C(4*n+2, 4*k-1) * a(k) * a(n+1-k). - _Sean A. Irvine_, Apr 01 2018

%F G.f. as a continued fraction: y(x) = x^3/(3 - x^4/(7 - x^4/(11 - x^4/(15 - ...)))) = 2*x^3/3! + 80*x^7/7! + 38400*x^11/11! + 77875200*x^15/15! + .... See Ellis. - _Peter Bala_, Jun 03 2019

%F y = Sum_{n>0} a(n) * x^(4*n-1)/(4*n-1)!. - _Michael Somos_, Mar 10 2020

%e y = 1/3*x^3 + 1/63*x^7 + 2/2079*x^11 + 13/218295*x^15 + 46/12442815*x^19 + ... - _Michael Somos_, Mar 10 2020

%t a[ n_] := a[n] = Which[n<1, 0, n==1, 2, True, Sum[ Binomial[4 n - 2, 4 k - 1] a[k] a[n - k], {k, n - 1}]]; (* _Michael Somos_, Mar 10 2020 *)

%o (PARI) {a(n) = my(v); if( n<1, 0, v=vector(n, m, 2); for(m=2, n, v[m] = sum(k=1, m-1, binomial(4*m-2, 4*k-1) * v[k] * v[m-k])); v[n])}; /* _Michael Somos_, Mar 10 2020 */

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, tony(AT)mantis.co.uk (Tony Lezard)