A257544 E.g.f.: 1/(1 - x/(1 - (x^2/2!)/(1 - (x^3/3!)/(1 - (x^4/4!)/(1 - (x^5/5!)/(1 -... (x^n/n!)/(1 -...))))))), a continued fraction.

1, 1, 2, 9, 48, 330, 2760, 26670, 295680, 3686760, 51067800, 778239000, 12937478400, 232997965200, 4518981066600, 93905637625800, 2081477759961600, 49020847858382400, 1222400311155223200, 32175639432544608000, 891491725439101987200, 25935626681521134177600, 790459388855441635730400

Offset

0,3

Comments

Limit n * a(n)/a(n+1) = 0.721838205529056...

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

a(n) ~ c * d^n * n!, where d = 1.385351997636466377154331548358490992212171811742252828163104808424069377... and c = 0.54049290018070557137173379570192477214723097220914603426642758802700306... - Vaclav Kotesovec, Aug 24 2017

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 330*x^5/5! +...
where
A(x) = 1 / (1 - x / (1 - (x^2/2!) / (1 - (x^3/3!) / (1 - (x^4/4!) / (1 - (x^5/5!) / (1 - (x^6/6!) / (1 - (x^7/7!) / (1 -...))))))))).

Mathematica

nmax = 30; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^Range[nmax + 1]/Range[nmax + 1]!]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 24 2017 *)

Prog

(PARI) {a(n)=local(CF=1+x*O(x^n), M=n+1); for(k=0, M, CF=1/(1-x^(M-k+1)/(M-k+1)!*CF)); n!*polcoeff(CF, n, x)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A358264 A246759 A191005 * A295944 A356632 A224140

Adjacent sequences: A257541 A257542 A257543 * A257545 A257546 A257547

Keyword

nonn

Author

Paul D. Hanna, Apr 28 2015

Status

approved