A346427 E.g.f.: -log(1 - log(1 + x) * exp(x)).

0, 1, 2, 7, 29, 183, 1319, 12122, 124802, 1508581, 20150509, 302637564, 4960500764, 89164162579, 1730245993111, 36241995276276, 812108432244304, 19430625834864633, 493622198791114665, 13283773364613034324, 377224137563670860492, 11278211794764786428831

Offset: 0

Author

Ilya Gutkovskiy, Jul 18 2021

Keywords

• nonn

Links

• Index entries for sequences related to logarithmic numbers

Crossrefs

Cf. A002741, A009321, A009324, A298374, A345454.

Programs

Mathematica

nmax = 21; CoefficientList[Series[-Log[1 - Log[1 + x] Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
A002741[n_] := A002741[n] = n! Sum[(-1)^k/((n - k) k!), {k, 0, n - 1}]; a[0] = 0; a[n_] := a[n] = -(-1)^n A002741[n] - (1/n) Sum[(-1)^(n - k) Binomial[n, k] A002741[n - k] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 21}]

PARI

my(x='x+O('x^25)); concat(0, Vec(serlaplace(-log(1 - log(1+x) * exp(x))))) \\ Michel Marcus, Jul 19 2021

Formula

a(0) = 0; a(n) = -(-1)^n * A002741(n) - (1/n) * Sum_{k=1..n-1} (-1)^(n-k) * binomial(n,k) * A002741(n-k) * k * a(k).