A355886 a(n) = n! * Sum_{k=1..n} floor(n/k)/k!.
1, 5, 22, 125, 746, 5677, 44780, 420401, 4206970, 47543141, 562891352, 7573655905, 104684547566, 1596368400005, 25482043382476, 439969180782017, 7835163501390290, 151712475696833221, 3004182138648663200, 63854641556089628801, 1400563708969910620822
Offset: 1
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..448
- Vaclav Kotesovec, Plot of a(n)/(n!*n) for n = 1..10000
Programs
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Mathematica
Table[n! * Sum[Floor[n/k]/k!, {k,1,n}], {n,1,25}] (* Vaclav Kotesovec, Aug 11 2025 *) -
PARI
a(n) = n!*sum(k=1, n, n\k/k!);
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k)))/(1-x))) -
PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, exp(x^k)-1)/(1-x))) -
PARI
a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/d!)); \\ Seiichi Manyama, Aug 08 2022
Formula
E.g.f.: (1/(1-x)) * Sum_{k>0} x^k/(k! * (1 - x^k)).
E.g.f.: (1/(1-x)) * Sum_{k>0} (exp(x^k) - 1).
a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/d! = n! * Sum_{k=1..n} A057625(k)/k!. - Seiichi Manyama, Aug 08 2022
a(n) ~ A229837 * n! * n. - Vaclav Kotesovec, Aug 11 2025