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A305307 Expansion of e.g.f. 1/(1 - log(1 + x)/(1 - x)). 3
1, 1, 3, 17, 120, 1084, 11642, 146446, 2101656, 33958344, 609431232, 12033015840, 259163792016, 6047213451408, 151953760489008, 4091057804809104, 117485988199385088, 3584814699783432960, 115816462543697120640, 3949619921174717629056, 141780511159572486530304, 5344008726418981985707776 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(n)/n! is the invert transform of [1, 1 - 1/2, 1 - 1/2 + 1/3, 1 - 1/2 + 1/3 - 1/4, 1 - 1/2 + 1/3 - 1/4 + 1/5, ...].
LINKS
N. J. A. Sloane, Transforms
FORMULA
E.g.f.: 1/(1 - Sum_{k>=1} (A058313(k)/A058312(k))*x^k).
a(n) ~ n! * (2 - LambertW(exp(2))) / ((1 + 1/LambertW(exp(2))) * (LambertW(exp(2)) - 1)^(n+1)). - Vaclav Kotesovec, Aug 08 2021
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1084*x^5/5! + 11642*x^6/6! + ...
MAPLE
g:= proc(n) g(n):= `if`(n=1, 0, g(n-1))-(-1)^n/n end:
b:= proc(n) option remember; `if`(n=0, 1,
add(g(j)*b(n-j), j=1..n))
end:
a:= n-> b(n)*n!:
seq(a(n), n=0..20); # Alois P. Heinz, May 29 2018
MATHEMATICA
nmax = 21; CoefficientList[Series[1/(1 - Log[1 + x]/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[1/(1 - Sum[Sum[(-1)^(j + 1)/j, {j, 1, k}] x^k , {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[((-1)^(k + 1) LerchPhi[-1, 1, k + 1] + Log[2]) a[n - k], {k, 1, n}]; Table[n! a[n], {n, 0, 21}]
CROSSREFS
Sequence in context: A121572 A340993 A249924 * A074543 A216314 A330803
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 29 2018
STATUS
approved

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Last modified March 28 07:48 EDT 2024. Contains 371235 sequences. (Running on oeis4.)