A301385 O.g.f. A(x) satisfies: A(x) = x * (1 + 3*x*A'(x)) / (1 + x*A'(x)).

1, 2, 6, 22, 94, 474, 2974, 24630, 271710, 3799570, 63378806, 1208997078, 25736584670, 602485683530, 15356903176110, 423032451327510, 12518043710674878, 395909541133928226, 13325077980379707238, 475466006418129789206, 17926802213221278261726, 712095926927360739006522, 29722097317161256669118142, 1300445348644716445771904502

Offset

1,2

Comments

If G(x) = x*(1 + r*x*G'(x)) / (1 + x*G'(x)), then G(x) has negative coefficients if r < t, and consists entirely of nonnegative coefficients if r > t, where t = 2.8453449032025472172778433620905570976610361149... (A301389).

O.g.f. equals the logarithm of the e.g.f. of A301386.

The e.g.f. G(x) of A301386 satisfies: [x^n] G(x)^(-n) = (2*n - 3) * [x^(n-1)] G(x)^(-n) for n>=1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

O.g.f. A(x) satisfies: [x^n] exp( -n * A(x) ) = (2*n - 3) * [x^(n-1)] exp( -n * A(x) ) for n>=1.

From Vaclav Kotesovec, Mar 20 2018: (Start)

a(n) ~ c * 2^n * n! / n^2, where c = 0.0618315205229178422646235585879521967924163...

a(n) ~ c * 2^n * n^(n - 3/2) / exp(n), where c = 0.15498863760617284891466946263730170095444214... (End)

Example

G.f.: A(x) = x + 2*x^2 + 6*x^3 + 22*x^4 + 94*x^5 + 474*x^6 + 2974*x^7 + 24630*x^8 + 271710*x^9 + 3799570*x^10 + ...
where
A(x) = x*(1 + 3*x*A'(x)) / (1 + x*A'(x)).
RELATED SERIES.
A'(x) = 1 + 4*x + 18*x^2 + 88*x^3 + 470*x^4 + 2844*x^5 + 20818*x^6 + 197040*x^7 + 2445390*x^8 + 37995700*x^9 + ...
exp(A(x)) = 1 + x + 5*x^2/2! + 49*x^3/3! + 745*x^4/4! + 16001*x^5/5! + 472621*x^6/6! + 19659025*x^7/7! + 1211940689*x^8/8! + ... + A301386*x^n/n! + ...

Mathematica

Rest[CoefficientList[AsymptoticDSolveValue[{A[x] == x*(1 + 3*x*A'[x])/(1 + x*A'[x]), A[1] == 1}, A[x], {x, 0, 20}], x]]  (* Requires Mathematica version 11.3 or later *) (* Vaclav Kotesovec, Mar 20 2018 *)

Prog

(PARI) {a(n) = my(A=x); for(i=0, n, A = x*(1 + 3*x*A')/(1 +x*A' +x*O(x^n)) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A301386, A301388, A301389, A300736, A300987.

Sequence in context: A049526 A187251 A193763 * A093793 A087959 A292096

Adjacent sequences: A301382 A301383 A301384 * A301386 A301387 A301388

Keyword

nonn

Author

Paul D. Hanna, Mar 20 2018

Status

approved