A216966 O.g.f.: 1/(1 - x/(1 - 2^3*x/(1 - 3^3*x/(1 - 4^3*x/(1 - 5^3*x/(1 - 6^3*x/(1 -...))))))), a continued fraction.

1, 1, 9, 297, 24273, 3976209, 1145032281, 530050022073, 369626762653857, 369614778179835681, 509880429246329788329, 940535818601273787325257, 2261104378216803649437779313, 6933711495845384616312688513329, 26630255658298074277771723491847161

Offset: 0

Author

Paul D. Hanna, Sep 20 2012

Keywords

• nonn

Comments

Compare to the continued fraction o.g.f. for the Euler numbers (A000364):

1/(1-x/(1-2^2*x/(1-3^2*x/(1-4^2*x/(1-5^2*x/(1-6^2*x/(1-...))))))).

From Vaclav Kotesovec, Sep 24 2020: (Start)

In general, if s>0 and g.f. = 1/(1 - x/(1 - 2^s*x/(1 - 3^s*x/(1 - 4^s*x/(1 - 5^s*x/(1 - 6^s*x/(1 -...))))))), a continued fraction, then

a(n,s) ~ c(s) * d(s)^n * (n!)^s / sqrt(n), where

d(s) = (2*s*Gamma(2/s) / Gamma(1/s)^2)^s

c(s) = sqrt(s*d(s)/(2*Pi)). (End)

Examples

			G.f.: A(x) = 1 + x + 9*x^2 + 297*x^3 + 24273*x^4 + 3976209*x^5 +...
		

Crossrefs

Cf. A000364, A227887, A337807, A337808, A337809.

Programs

Maple

T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else -(k - n - 1)^3 * T(n, k - 1) + T(n - 1, k) fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..14);  # Peter Luschny, Oct 02 2023

Mathematica

nmax = 20; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Range[nmax + 1]^3*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)

PARI

{a(n)=local(CF=1+x*O(x^n)); for(k=1, n, CF=1/(1-(n-k+1)^3*x*CF)); polcoeff(CF,n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f.: T(0), where T(k) = 1 - x*(k+1)^3/(x*(k+1)^3 -1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2013

a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 192 * sqrt(3) * Pi^3 / Gamma(1/3)^9 = 1.450930901627203932388423902788627... and c = 12 * sqrt(2) * 3^(1/4) * Pi / Gamma(1/3)^(9/2) = sqrt(3*d/(2*Pi)) = 0.8323271443586650769764930497... - Vaclav Kotesovec, Aug 25 2017, updated Sep 23 2020