A330493
a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * tau(k), where tau = A000005.
Original entry on oeis.org
1, 3, 12, 70, 492, 4298, 43894, 514666, 6830888, 101473632, 1664125944, 29858266392, 582481147440, 12281821373040, 278257595964576, 6739505703156192, 173785740554811264, 4754455742416944000, 137571331202872821504, 4197696814883284962048
Offset: 1
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Table[Sum[(-1)^(n-k) * StirlingS1[n, k] * (k-1)! * DivisorSigma[0, k], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[-Sum[Log[1 - Log[1/(1 - x)]^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]
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a(n) = sum(k=1, n, (-1)^(n-k)*stirling(n, k, 1)*(k-1)!*numdiv(k)); \\ Michel Marcus, Dec 16 2019
A330494
a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * sigma(k), where sigma = A000203.
Original entry on oeis.org
1, 4, 19, 129, 1018, 9912, 111074, 1416398, 20295208, 323437728, 5657339928, 107765338920, 2223272444976, 49399021063584, 1175549092374672, 29822113966614768, 803485297880792064, 22917198585269729664, 689927737384840662144, 21861972842959846530432
Offset: 1
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Table[Sum[(-1)^(n-k) * StirlingS1[n, k] * (k-1)! * DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]
nmax = 20; Rest[CoefficientList[Series[Sum[Log[1/(1 - x)]^k/(k (1 - Log[1/(1 - x)]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!]
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a(n) = sum(k=1, n, (-1)^(n-k)*stirling(n, k, 1)*(k-1)!*sigma(k)); \\ Michel Marcus, Dec 16 2019
A330450
Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)^2).
Original entry on oeis.org
1, 4, 7, 55, -162, 4100, -49030, 779914, -11928008, 198650880, -3538477560, 70414760136, -1571134087824, 38788172175072, -1028732373217200, 28631225505910224, -826097667884640768, 24664145505337921920, -765245501125015575168, 24841409653689047496576
Offset: 1
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nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[2, k], {k, 1, n}], {n, 1, 20}]
Showing 1-3 of 3 results.