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;
A356394
Expansion of e.g.f. ( Product_{k>0} (1+x^k)^k )^(1/(1-x)).
Original entry on oeis.org
1, 1, 6, 51, 452, 5210, 68514, 1032906, 17352320, 323948376, 6594052680, 145585638000, 3461441121192, 88092914635128, 2388119359650192, 68667743686492440, 2086307088847714560, 66762608893508354880, 2243693428523140377024, 78982154604162553529664
Offset: 0
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nmax = 20; CoefficientList[Series[Product[(1+x^k)^k, {k, 1, nmax}]^(1/(1-x)), {x, 0, nmax}], x] * Range[0,nmax]! (* Vaclav Kotesovec, Aug 07 2022 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^(1/(1-x))))
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a356391(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/k);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356391(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)))
A356565
Expansion of e.g.f. ( Product_{k>0} (1+x^k) )^x.
Original entry on oeis.org
1, 0, 2, 3, 44, 90, 2034, 9240, 168944, 951048, 24042600, 185387400, 4411634952, 44020650960, 1166597641104, 14101322278680, 399099955203840, 5522583764698560, 169123038510919104, 2779010889700890240, 87888034148774728320, 1637061268780618450560
Offset: 0
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nmax = 20; CoefficientList[Series[Product[(1 + x^k)^x, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 17 2022 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+x^k)^x))
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a000593(n) = sumdiv(n, d, (-1)^(n/d+1)*d);
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*a000593(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-4 of 4 results.