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A180515 E.g.f. A(x) satisfies A''(x) = 2*A(x)^3 + x*A(x) + 1. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

The exponential generating function A(x) = x^2/+x^5/40 +11*x^8/2240+... satisfies the PainlevĂ© 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

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Last modified May 17 13:48 EDT 2022. Contains 353746 sequences. (Running on oeis4.)