A352013
a(n) = Sum_{d|n} (-1)^(n/d+1) * (n-1)!/(d-1)!.
Original entry on oeis.org
1, 0, 3, -11, 25, -59, 721, -10919, 60481, -15119, 3628801, -93471839, 479001601, -8648639, 134399865601, -2833553923199, 20922789888001, -174888473759999, 6402373705728001, -228084898487846399, 3652732042831872001, -14079294028799
Offset: 1
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restart;
f:= proc(n) local d;
add((-1)^(n/d + 1) * (n-1)!/(d-1)!, d = numtheory:-divisors(n))
end proc:
map(f, [$1..30]); # Robert Israel, Nov 14 2024
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a[n_] := DivisorSum[n, (-1)^(n/#+1) * (n-1)!/(#-1)! &]; Array[a, 22] (* Amiram Eldar, Aug 30 2023 *)
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a(n) = sumdiv(n, d, (-1)^(n/d+1)*(n-1)!/(d-1)!);
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my(N=40, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k)/k!)))
A356402
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k!) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 3, 16, 86, 626, 5267, 50793, 543279, 6544805, 86503762, 1242678141, 19259416827, 321457169151, 5736414618209, 108931865485750, 2191495621647324, 46604972526167314, 1043844453093239627, 24555321244430950299, 605239630722584461955, 15600222966916650541099
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k!))^(1/(1-x))))
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a356401(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(d+1)/(d*(k/d)!)));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356401(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A345757
E.g.f.: Product_{k>=1} (1 + (exp(x) - 1)^k)^(1/k!).
Original entry on oeis.org
1, 1, 2, 8, 34, 137, 614, 3754, 25449, 82747, -1523792, -34833005, -335209288, 194665837, 59685834069, 715582325511, -10186972407657, -584687267399246, -10975484551366964, 8845584310341044, 8145484883568515927, 330326712925212377392, 7816903733527799885488
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+(exp(x)-1)^k)^(1/k!))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, (-1)^(k+1)*(exp((exp(x)-1)^k)-1)/k))))
A345870
Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/k!).
Original entry on oeis.org
1, 2, 6, 26, 126, 742, 5166, 40462, 351742, 3458470, 37425406, 440788702, 5633316574, 77379974518, 1140707915262, 18053421105742, 302414295475134, 5364631473148614, 100769601500958078, 1988246969908681278, 41179474537324087454, 896909297854081874454
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, ((1+x^k)/(1-x^k))^(1/k!))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sum(k=0, N\2, (exp(x^(2*k+1))-1)/(2*k+1)))))
A345871
Expansion of e.g.f. Product_{k>=1} ((1 + x^k)/(1 - x^k))^(1/(2*k!)).
Original entry on oeis.org
1, 1, 2, 7, 23, 116, 707, 4775, 34092, 326723, 3255927, 35131570, 404387205, 5178352921, 72399248378, 1124473250815, 17051538263075, 287692287405292, 5225129815132463, 97469631563584567, 1891891950093538380, 40846072052629411027, 895935864764993940483
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, ((1+x^k)/(1-x^k))^(1/(2*k!)))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\2, (exp(x^(2*k+1))-1)/(2*k+1)))))
A352005
Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k))^(1/prime(k)!).
Original entry on oeis.org
1, 0, 1, 1, -3, 11, -5, -83, -2919, 18838, 118371, 583826, -27365327, -12780260, 405396069, 32646641041, -232690739007, 4816360930145, -46984166770283, -541620811734953, -49355727191815599, 907100235094018036, 10877428540752188625, 139350853273096742762
Offset: 0
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my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(isprime(k)/k!))))
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, isprime(d)*(-1)^(k/d+1)*(k-1)!/(d-1)!)*x^k/k!))))
Showing 1-6 of 6 results.
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