A338193 E.g.f. A(x) satisfies: A(x) = 1 + Integral (x/A(x))' / (x/A(x)^2)' dx.

1, 1, 2, 10, 100, 1556, 33016, 888952, 29035280, 1115554960, 49300214176, 2463859486496, 137403573562432, 8459184183342400, 569861708317147520, 41697486853043633536, 3293243089832744386816, 279234174032057551630592, 25299360271944290704683520

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Formula

a(n) ~ (1 + sqrt(2))^(2*n - 3/2) * n^(n-2) / (2^(3/4) * exp(n - 1 + 1/sqrt(2))).

E.g.f.: (8*x^2 * (x - 3 + sqrt(1 - 6*x + x^2))^3 / (3*x - 1 + sqrt(1 - 6*x + x^2)))^(1/4) * exp((1 + x - sqrt(1 - 6*x + x^2))/4)/2.

Mathematica

nmax = 20; A = 1; Do[A = 1 + Integrate[D[x/A, x]/D[x/A^2, x], x] + O[x]^nmax, nmax]; CoefficientList[A, x] * Range[0, nmax - 1]!
nmax = 20; FullSimplify[CoefficientList[Series[(8*x^2 * (x - 3 + Sqrt[1 - 6*x + x^2])^3 / (3*x - 1 + Sqrt[1 - 6*x + x^2]))^(1/4) * E^((1 + x - Sqrt[1 - 6*x + x^2])/4)/2, {x, 0, nmax}], x] * Range[0, nmax]!]

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = 1 + intformal( (x/A)'/(x/A^2 +x*O(x^n))' ); ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A302701, A303064, A338163, A338187, A338188, A338194.

Sequence in context: A228990 A277468 A099826 * A063959 A101686 A324241

Adjacent sequences: A338190 A338191 A338192 * A338194 A338195 A338196

Keyword

nonn

Author

Vaclav Kotesovec, Oct 15 2020

Status

approved