OFFSET
0,3
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f. A(x) satisfies: A(x) = A(-x/(1-4*x))/sqrt(1-4*x).
Eigenvector: a(n) = Sum_{k=0..floor(n/2)} a(k)*(4*k+1)*binomial(2*n+1,n-2*k)/(2*n+1) for n>=0, with a(0)=1.
From Peter Bala, Sep 16 2023: (Start)
It appears that the g.f. A(x) satisfies A(x^2) = 1/(1 + x)*A(x/(1 + x)^2).
If true, then F(x) := A(x^2) satisfies the functional equation F(x^2) = 1/(1 + x^2)*F(x/(1 + x^2)). Compare with Landen's transformation K(x^2) = 1/(1 + x^2)*K(2*x/(1 + x^2)) (for 0 <= x < 1) for the complete elliptic integral of the first kind K(x) = Integral_{t = 0..Pi/2} dt/sqrt(1 - x^2*sin^2(t)). (End)
PROG
(PARI) {a(n)=if(n==0, 1, sum(k=0, n\2, a(k)*(4*k+1)*binomial(2*n+1, n-2*k)/(2*n+1)))}
(PARI) seq(n) = {my(a=vector(n+1)); a[1]=1; for(n=1, n, a[1+n] = sum(k=0, n\2, a[1+k]*(4*k+1)*binomial(2*n+1, n-2*k))/(2*n+1)); a} \\ Andrew Howroyd, Sep 19 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 10 2006
STATUS
approved