A355886
a(n) = n! * Sum_{k=1..n} floor(n/k)/k!.
Original entry on oeis.org
1, 5, 22, 125, 746, 5677, 44780, 420401, 4206970, 47543141, 562891352, 7573655905, 104684547566, 1596368400005, 25482043382476, 439969180782017, 7835163501390290, 151712475696833221, 3004182138648663200, 63854641556089628801, 1400563708969910620822
Offset: 1
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Table[n! * Sum[Floor[n/k]/k!, {k,1,n}], {n,1,25}] (* Vaclav Kotesovec, Aug 11 2025 *)
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a(n) = n!*sum(k=1, n, n\k/k!);
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k/(k!*(1-x^k)))/(1-x)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, exp(x^k)-1)/(1-x)))
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a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/d!)); \\ Seiichi Manyama, Aug 08 2022
A356025
Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k)^(1/k!) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 5, 28, 206, 1786, 18347, 212745, 2773927, 39901109, 628298992, 10725440221, 197349522471, 3888090474399, 81659016005387, 1820049574958950, 42895622543757084, 1065460090285463634, 27811791343693345811, 760920657403831436463
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace((1/prod(k=1, N, (1-x^k)^(1/k!)))^(1/(1-x))))
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a356009(n) = n!*sum(k=1, n, sumdiv(k, 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, a356009(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356406
a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d * (k/d)^d).
Original entry on oeis.org
1, 4, 16, 79, 443, 2968, 22216, 189698, 1792402, 18745036, 213452996, 2653142952, 35448861576, 509724975264, 7824794618208, 128006170541328, 2217950478978576, 40686737647774368, 785852762719168992, 15974195890305405696, 340376906088298319616
Offset: 1
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a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)^d)));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k/k))/(1-x)))
A356407
a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d * ((k/d)!)^d).
Original entry on oeis.org
1, 4, 15, 70, 375, 2411, 17598, 146490, 1359291, 13978597, 157393368, 1929989029, 25568858978, 364288345409, 5551537358188, 90142504077194, 1553345359200299, 28317316174307405, 544431381017568696, 11010510372888267555, 233653645911730002976
Offset: 1
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a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!^d)));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k/k!))/(1-x)))
A356401
a(n) = n! * Sum_{k=1..n} Sum_{d|k} (-1)^(d+1)/(d * (k/d)!).
Original entry on oeis.org
1, 2, 9, 25, 150, 841, 6608, 41945, 437986, 4364741, 51640952, 526219585, 7319856206, 102469338245, 1671439939276, 23909485105217, 427384036676690, 7518024186420421, 149244833247716000, 2756811766466473601, 61545779138627817622, 1354007126970517958885
Offset: 1
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a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(d+1)/(d*(k/d)!)));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, (-1)^k*(exp(x^k)-1)/k)/(1-x)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, log(1+x^k)/k!)/(1-x)))
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)))
A354338
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} 1/(d * (k/d)!) )/(n-k)!.
Original entry on oeis.org
1, 4, 12, 41, 145, 742, 3962, 27659, 215131, 1996356, 17300360, 218809109, 2421142269, 31105286682, 427776526574, 6964677268087, 97708052695959, 1856379196278120, 30362097934331500, 606395795174882161, 12016899266310773097, 261771941015999635310
Offset: 1
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a087906(n) = n!*sumdiv(n, d, 1/(d*(n/d)!));
a(n) = sum(k=1, n, a087906(k)*binomial(n, k));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=1, N, (exp(x^k)-1)/k)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, log(1-x^k)/k!)))
Showing 1-7 of 7 results.
