A352693
Expansion of e.g.f. 1 / (1 - Sum_{k>=1} sigma_2(k) * x^k/k!).
Original entry on oeis.org
1, 1, 7, 46, 455, 5406, 78172, 1312116, 25214479, 544777183, 13080808752, 345471545728, 9953804592152, 310687941345796, 10443489230611052, 376122782541917166, 14449157656748079247, 589772212576633845886, 25488817336672959449725
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-sum(k=1, N, sigma(k, 2)*x^k/k!))))
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a(n) = if(n==0, 1, sum(k=1, n, sigma(k, 2)*binomial(n, k)*a(n-k)));
A307242
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1)*sigma_2(k+1)*a(n-k), where sigma_2() is the sum of squares of divisors (A001157).
Original entry on oeis.org
1, 5, 15, 46, 159, 570, 2036, 7208, 25400, 89456, 315335, 1112286, 3923867, 13841052, 48818892, 172186234, 607314043, 2142064478, 7555322206, 26648517536, 93992371863, 331521717928, 1169314641890, 4124305724658, 14546896171716, 51308559972146, 180971133233105, 638305788168090
Offset: 0
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a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) DivisorSigma[2, k + 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]
nmax = 27; CoefficientList[Series[-x/Sum[k^2 (-x)^k/(1 - (-x)^k), {k, 1, nmax + 1}], {x, 0, nmax}], x]
nmax = 27; CoefficientList[Series[1/D[Log[Product[(1 - (-x)^k)^k, {k, 1, nmax + 1}]], x], {x, 0, nmax}], x]
A321190
a(n) = [x^n] 1/(1 - Sum_{k>=1} k^n*x^k/(1 - x^k)).
Original entry on oeis.org
1, 1, 6, 47, 778, 25476, 1752936, 242632397, 70015221566, 41446777283255, 49999934258165654, 125272856707074638221, 641938223803783115191706, 6731818441446626626586172740, 146378489075644780343627471981694, 6505906463580477520696075719916583118
Offset: 0
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seq(coeff(series((1-add(k^n*x^k/(1-x^k),k=1..n))^(-1),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 29 2018
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Table[SeriesCoefficient[1/(1 - Sum[k^n x^k/(1 - x^k), {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[DivisorSigma[n, k] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 15}]
Table[SeriesCoefficient[1/(1 - Sum[Sum[j^n x^(i j), {j, 1, n}], {i, 1, n}]), {x, 0, n}], {n, 0, 15}]
A352839
Expansion of g.f. 1/(1 - Sum_{k>=1} sigma_k(k) * x^k).
Original entry on oeis.org
1, 1, 6, 39, 370, 4132, 59288, 990705, 19577018, 439550259, 11142216938, 313147651821, 9680830606850, 325944181383936, 11875777329091878, 465292113335910106, 19507503314546762246, 871248546067010133794, 41295079536653463057146
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(1/(1-sum(k=1, N, sigma(k, k)*x^k)))
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a(n) = if(n==0, 1, sum(k=1, n, sigma(k, k)*a(n-k)));
Showing 1-4 of 4 results.