OFFSET
0,4
FORMULA
a(n) = sum(m=0..(n-1)/2, a(n-2*m-1) * binomial(n/2-1,m) * 2^(2*m)), n>0, a(0)=1.
G.f.: Sum_{n>=0} x^n / Product_{k=0..n} sqrt(1 - 4*k*x^2). - Paul D. Hanna, Nov 30 2014
G.f. A(x) satisfies: A(x/sqrt(1+4*x^2)) = 1 + x*A(x). - Paul D. Hanna, Nov 30 2014
EXAMPLE
G.f.: A(x) = = 1 + x + x^2 + 3*x^3 + 7*x^4 + 19*x^5 + 59*x^6 + 179*x^7 + ...
such that A(x) = 1 + G(x)*A(G(x)) where
G(x) = x/sqrt(1-4*x^2) = x + 2*x^3 + 6*x^5 + 20*x^7 + 70*x^9 + 252*x^11 + ... .
The g.f. also equals the series
A(x) = 1 + x/sqrt(1-4*x^2) + x^2/sqrt((1-4*x^2)*(1-8*x^2)) + x^3/sqrt((1-4*x^2)*(1-8*x^2)*(1-12*x^2)) + x^4/sqrt((1-4*x^2)*(1-8*x^2)*(1-12*x^2)*(1-16*x^2)) + ... .
PROG
(Maxima)
a(n):=if n=0 then 1 else sum(a(n-2*m-1)*binomial(n/2-1, m)*2^(2*m), m, 0, (n-1)/2);
(PARI) {a(n)=polcoeff( sum(m=0, n, x^m / prod(k=0, m, sqrt(1 - 4*k*x^2 +x*O(x^n)) )), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 30 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Nov 28 2011
EXTENSIONS
Name corrected slightly by Paul D. Hanna, Nov 30 2014
STATUS
approved