A356389
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d + 1) ) /k.
Original entry on oeis.org
1, 2, 10, 34, 218, 1308, 10596, 74688, 793152, 7931520, 94504320, 1054218240, 14662840320, 205279764480, 3427909632000, 50923531008000, 907545606912000, 16335820924416000, 323185344975360000, 6220416698689536000, 140360358705186816000, 3087927891514109952000
Offset: 1
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Table[n! * Sum[Sum[-(-1)^(k/d), {d, Divisors[k]}]/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
Table[n! * Sum[(2*DivisorSigma[0, 2*k] - 3*DivisorSigma[0, k])/k, {k, 1, n}], {n, 1, 25}] (* Vaclav Kotesovec, Aug 07 2022 *)
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a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1))/k);
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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)))
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)))
A356393
Expansion of e.g.f. ( Product_{k>0} (1+x^k) )^(1/(1-x)).
Original entry on oeis.org
1, 1, 4, 27, 188, 1730, 18234, 220206, 2958416, 44470296, 729675720, 13002636240, 249986061192, 5154030469848, 113360272804128, 2648908519611480, 65477559553098240, 1707034986277780800, 46798324479957887424, 1345365460101611611584
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+x^k)^(1/(1-x))))
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a356390(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356390(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A354507
a(n) = n! * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d+1) * d )/(k * (n-k)!).
Original entry on oeis.org
1, 3, 14, 48, 269, 1615, 12662, 73528, 836817, 8476243, 99348534, 948849176, 13193115597, 177346261391, 3684976294222, 45021819481808, 734808219625345, 13524660020400771, 290452222949307070, 4639956700466396256, 128621330002689008237, 2735863084773695212719
Offset: 1
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a(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d)/(k*(n-k)!));
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*sum(k=1, N, (-x)^k/(k*(1-x^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))))
Showing 1-4 of 4 results.