A337809 O.g.f.: 1/(1 - x/(1 - 2^7*x/(1 - 3^7*x/(1 - 4^7*x/(1 - 5^7*x/(1 - 6^7*x/(1 -...))))))), a continued fraction.

1, 1, 129, 296577, 5273061633, 456296857756929, 143521873041157216641, 134210828762693919568092033, 322179101908965036802512977670657, 1775143826590061506939568896182460951041, 20554318541749460884980441781629250054049026689

Offset: 0

Author

Vaclav Kotesovec, Sep 23 2020

Keywords

• nonn

Comments

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 and c(s) = sqrt(s*d(s)/(2*Pi)). - Vaclav Kotesovec, Sep 24 2020

Crossrefs

Cf. A001147, A000364, A216966, A227887, A337807, A337808.

Programs

Mathematica

nmax = 15; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[Range[nmax + 1]^7*x]], {x, 0, nmax}], x]

Formula

a(n) ~ c * d^n * (n!)^7 / sqrt(n), where

d = 14^7 * Gamma(2/7)^7 / Gamma(1/7)^14 = 1.2151675804792498774003050188354949771364793751019885755525736...

c = sqrt(7*d/(2*Pi)) = 1.1635288951410008357326423559026931516828251494058147648...