A354862
a(n) = n! * Sum_{d|n} (n/d)! / d!.
Original entry on oeis.org
1, 5, 37, 601, 14401, 520801, 25401601, 1626189601, 131682257281, 13168407228481, 1593350922240001, 229442707280223361, 38775788043632640001, 7600054676241325858561, 1710012252750418295078401, 437763137119219420513804801, 126513546505547170185216000001
Offset: 1
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a[n_] := n! * DivisorSum[n, (n/#)! / #! &]; Array[a, 17] (* Amiram Eldar, Aug 30 2023 *)
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a(n) = n!*sumdiv(n, d, (n/d)!/d!);
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k!*(exp(x^k)-1))))
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from math import factorial
from sympy import divisors
def A354862(n):
f = factorial(n)
return sum(f*(a := factorial(n//d))//(b:= factorial(d)) + (f*b//a if d**2 < n else 0) for d in divisors(n,generator=True) if d**2 <= n) # Chai Wah Wu, Jun 09 2022
A356459
a(n) = n! * Sum_{k=1..n} Sum_{d|k} d/(k/d)!.
Original entry on oeis.org
1, 7, 40, 281, 2006, 17677, 159020, 1678721, 18555850, 230978981, 2979853592, 43323807265, 644160764846, 10543905398405, 178896116995276, 3284281839169217, 61879477543508690, 1264313089711322821, 26333205612282941600, 588074615109602665601
Offset: 1
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Table[n! * Sum[Sum[d/(k/d)!, {d,Divisors[k]}], {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, Aug 11 2025 *)
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a(n) = n!*sum(k=1, n, sumdiv(k, d, d/(k/d)!));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*(exp(x^k)-1))/(1-x)))
A356460
Expansion of e.g.f. Product_{k>0} B(x^k)^k where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
Original entry on oeis.org
1, 1, 6, 35, 303, 2772, 32903, 410335, 6051692, 95183187, 1675869175, 31437027030, 644157830077, 13976891765137, 325719071472590, 8007861177420275, 208953947981129027, 5725964099963426924, 165258064179632753563, 4987477844227598529047
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1)^k)))
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a354863(n) = n!*sumdiv(n, d, n/d/d!);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354863(j)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-3 of 3 results.