OFFSET
1,1
REFERENCES
G. N. Watson, Bessel Functions, p. 2.
LINKS
H. G. Ellis, Continued fraction solutions of the general Riccati differential equation, Rocky Mountain Journal of Mathematics, Vol. 4, Number 2, 353-356, Spring 1974.
FORMULA
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).
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
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
y = Sum_{n>0} a(n) * x^(4*n-1)/(4*n-1)!. - Michael Somos, Mar 10 2020
EXAMPLE
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
MATHEMATICA
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 *)
PROG
(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 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, tony(AT)mantis.co.uk (Tony Lezard)
STATUS
approved