A357296 Expansion of e.g.f. Sum_{k>0} x^k / (k! * (1 - x^k/k)).

1, 3, 7, 31, 121, 851, 5041, 43261, 369601, 3748249, 39916801, 490801081, 6227020801, 87861842641, 1310800947457, 21018206008801, 355687428096001, 6419518510204801, 121645100408832001, 2435836129700029057, 51102829650622464001, 1124549558817839481601

Offset

1,2

Links

Table of n, a(n) for n=1..22.

Formula

a(n) = n! * Sum_{d|n} 1 / (d^(n/d-1) * d!).

If p is prime, a(p) = 1 + p! = A038507(p).

Mathematica

a[n_] := n! * DivisorSum[n, 1/(#^(n/#-1) * #!) &]; Array[a, 20] (* Amiram Eldar, Jul 31 2023 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k/k)))))
(PARI) a(n) = n!*sumdiv(n, d, 1/(d^(n/d-1)*d!));

Crossrefs

Cf. A038507, A057625, A327578, A354891.

Sequence in context: A306831 A105765 A061095 * A103901 A089162 A016047

Adjacent sequences: A357293 A357294 A357295 * A357297 A357298 A357299

Keyword

nonn

Author

Seiichi Manyama, Feb 23 2023

Status

approved