A356392
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^(1/k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 3, 17, 99, 769, 6877, 70769, 807321, 10366037, 145721531, 2226927405, 36741898267, 651709348653, 12352436747141, 249152882935829, 5320544034698353, 120008265471779529, 2850195632804141203, 71058458112629765449, 1855470903727083981651
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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a356389(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356389(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A356390
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) * d ) /k.
Original entry on oeis.org
1, 3, 17, 74, 514, 3564, 30708, 250704, 2780496, 29982240, 373350240, 4639870080, 67024333440, 988156834560, 16914631507200, 271941778483200, 4999620452198400, 94617104704819200, 1925772463506124800, 39245319872575488000, 902004581585737728000
Offset: 1
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Table[n! * Sum[Sum[(-1)^(k/d + 1)*d, {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
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a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/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, log(1+x^k))/(1-x)))
A356391
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) * d^2 ) /k.
Original entry on oeis.org
1, 5, 35, 206, 1654, 13524, 130668, 1262064, 15027696, 178581600, 2407111200, 33276182400, 514020643200, 8130342124800, 144621487584000, 2537556118272000, 49206063078144000, 982811803276800000, 20991083543732736000, 454612169591580672000, 10763306565511514112000
Offset: 1
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Table[n! * Sum[Sum[(-1)^(k/d + 1)*d^2, {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Aug 07 2022 *)
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a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/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)^2))/(1-x)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, k*log(1+x^k))/(1-x)))
A354506
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) )/(k * (n-k)!).
Original entry on oeis.org
1, 2, 7, 14, 63, 284, 2385, 3940, 87717, 940126, 12743267, 30055618, 562302323, 9005878920, 423435780989, 2080544097000, 24457758561001, 444510436079706, 17533073308723423, 46973556239255702, 7501223613055891783, 178483805340054632084, 4396051786608296882889, -31788150263554644516724
Offset: 1
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a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/(k*(n-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 1-4 of 4 results.