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a(n) = Sum_{p|n, p prime} (-1)^(n/p+1) * (n-1)!/(p-1)!.
+10
3
0, 1, 1, -6, 1, 60, 1, -5040, 20160, 347760, 1, -59875200, 1, 6218372160, 47221574400, -1307674368000, 1, 177843714048000, 1, -126713646259200000, 1219830034655232000, 51090928092415411200, 1, -38778025108327464960000, 25852016738884976640000
FORMULA
E.g.f.: Sum_{p prime} log(1+x^p)/p!.
MATHEMATICA
a[n_] := Sum[(-1)^(n/p + 1)*(n - 1)!/(p - 1)!, {p, FactorInteger[n][[;; , 1]]}]; a[1] = 0; Array[a, 25] (* Amiram Eldar, Oct 04 2023 *)
PROG
(PARI) a(n) = sumdiv(n, d, isprime(d)*(-1)^(n/d+1)*(n-1)!/(d-1)!);
(PARI) my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=1, N, isprime(k)*log(1+x^k)/k!))))
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(d+1)/(d * (k/d)!) )/(n-k)!.
+10
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1, 2, 6, 5, 5, -8, 560, -5997, -14765, 176826, 5206410, -42491623, -427057527, -412183484, 147180377804, -569782989113, -8367671807033, -119681999820906, 4440973420854454, -121033449284728099, 49772248126885197, 36615485147317407728, 1696495197400394891912
FORMULA
a(n) = Sum_{k=1..n} A352013(k) * binomial(n,k). E.g.f.: -exp(x) * Sum_{k>0} (-1)^k * (exp(x^k) - 1)/k.
E.g.f.: exp(x) * Sum_{k>0} log(1+x^k)/k!.
PROG
(PARI) a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(d+1)/(d*(k/d)!))/(n-k)!);
(PARI) a352013(n) = sumdiv(n, d, (-1)^(n/d+1)*(n-1)!/(d-1)!);
a(n) = sum(k=1, n, a352013(k)*binomial(n, k));
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, (-1)^k*(exp(x^k)-1)/k)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, log(1+x^k)/k!)))
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k!) )^x.
+10
1
1, 0, 2, 0, 24, -55, 630, -2723, 30968, -294327, 3047320, -30255379, 387690732, -5565964391, 77090414492, -1114263777885, 18473122449616, -331776991760303, 6106973926830192, -112710455017397639, 2233663985151902860, -50049383051597936559
FORMULA
a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k * A352013(k-1) * binomial(n-1,k-1) * a(n-k).
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k!))^x))
(PARI) a352013(n) = (n-1)!*sumdiv(n, d, (-1)^(n/d+1)/(d-1)!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*a352013(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
a(n) = (n-1)! * Sum_{d|n} (-1)^(d+1) / (d-1)!.
+10
1
1, 0, 3, -1, 25, 59, 721, -841, 60481, 15119, 3628801, 12972959, 479001601, 8648639, 134399865601, -218205187201, 20922789888001, 174888473759999, 6402373705728001, -15205972772390401, 3652732042831872001, 14079294028799, 1124000727777607680001
FORMULA
E.g.f.: Sum_{k>0} (1 - exp(-x^k))/k.
E.g.f.: Sum_{k>0} (-1)^k * log(1-x^k)/k!.
If p is an odd prime, a(p) = 1 + (p-1)!.
MATHEMATICA
a[n_] := (n-1)! * DivisorSum[n, (-1)^(#+1)/(#-1)! &]; Array[a, 25] (* Amiram Eldar, Jul 03 2023 *)
PROG
(PARI) a(n) = (n-1)!*sumdiv(n, d, (-1)^(d+1)/(d-1)!);
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