A180515 E.g.f. A(x) satisfies A''(x) = 2*A(x)^3 + x*A(x) + 1.

0, 0, 1, 0, 0, 3, 0, 0, 198, 0, 0, 15390, 0, 0, 4611168, 0, 0, 1829539224, 0, 0, 1492247906784, 0, 0, 1669958449339824, 0, 0, 2955696363525356640, 0, 0, 7028088099915471491520, 0, 0, 23308039026983275082311680, 0, 0, 100343481973929775498656672000

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).

Links

Table of n, a(n) for n=0..35.

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

Sequence in context: A160537 A215516 A009133 * A009138 A175562 A319330

Adjacent sequences: A180512 A180513 A180514 * A180516 A180517 A180518

Keyword

nonn

Author

Vladimir Kruchinin, Jan 21 2011

Extensions

a(18)-a(35) from Andrew Howroyd, Apr 17 2021

Status

approved