OFFSET
0,6
COMMENTS
The exponential generating function A(x) = x^2/+x^5/40 +11*x^8/2240+... satisfies the Painlevé II equation A''(x) = 2*A(x)^3+x*A(x)+1. This is the case b=1 of the more general A''(x) = 2*A(x)^3+x*A(x)+b which has a solution a(0)=a(1)=0, a(2)=b/2, a(3)=a(4)=0 and, for n>4, a(n) = (2*A(n-2,3)+a(n-3)) / (n*(n-1)) where A(n,1)=a(n) and the components of A(n,k) with k>1 are recursively A(n,k) = sum_{i=0..n-k} a(i+1)*A(n-i-1,k-1).
PROG
(PARI) seq(n)={my(p=x^2/2+O(x^3)); for(n=1, n, p = intformal(intformal(2*p^3 + x*p + 1))); Vec(serlaplace(p), -serprec(p, x))} \\ Andrew Howroyd, Apr 17 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Jan 21 2011
EXTENSIONS
a(18)-a(35) from Andrew Howroyd, Apr 17 2021
STATUS
approved