A309473 a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

1, 0, 1, 1, 3, 11, 43, 195, 1063, 6395, 42371, 311883, 2501159, 21672355, 202544323, 2028522067, 21658255431, 245738583307, 2952508103651, 37440976938875, 499785548010759, 7005210659040979, 102862231664567651, 1579045889274408259, 25294106622048460903

Offset

0,5

Links

Table of n, a(n) for n=0..24.

Formula

E.g.f. A(x) satisfies: A'(x) = A(x)^2 - exp(-x).

From Vaclav Kotesovec, Jun 11 2020: (Start)

E.g.f.: exp(-x/2) * (BesselI(1, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2)) + (BesselI(0, 2) - BesselI(1, 2)) * BesselK(1, 2*exp(-x/2))) / ((BesselI(1, 2) - BesselI(0, 2)) * BesselK(0, 2*exp(-x/2)) + BesselI(0, 2*exp(-x/2)) * (BesselK(0, 2) + BesselK(1, 2))).

a(n) ~ n! / r^(n+1), where r = 1.4982609322383959128764444062824740935658895762... is the real root of the equation (BesselI(0, 2) - BesselI(1, 2)) * BesselK(0, 2*exp(-r/2)) = (BesselK(0, 2) + BesselK(1, 2)) * BesselI(0, 2*exp(-r/2)). (End)

Mathematica

a[n_] := a[n] = (-1)^n + Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
terms = 24; A[_] = 1; Do[A[x_] = Normal[Integrate[A[x]^2 - Exp[-x], x] + O[x]^(terms + 1)], terms]; CoefficientList[A[x], x] Range[0, terms]!

Crossrefs

Cf. A006677, A014304, A054687, A332776, A335441.

Sequence in context: A126282 A039627 A231497 * A349058 A051257 A135482

Adjacent sequences: A309470 A309471 A309472 * A309474 A309475 A309476

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 11 2020

Status

approved