A330354
Expansion of e.g.f. Sum_{k>=1} log(1 + x)^k / (k * (1 - log(1 + x)^k)).
Original entry on oeis.org
1, 2, 1, 21, -122, 1752, -21730, 309166, -4521032, 70344768, -1173530712, 21642745704, -448130571696, 10352684535840, -260101132095888, 6921279885508848, -191813249398678272, 5502934340821289088, -163695952380982280832, 5078687529186002247552
Offset: 1
Cf.
A000041,
A000203,
A002743,
A008275,
A038048,
A089064,
A306042,
A330351,
A330352,
A330353,
A330494.
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nmax = 20; CoefficientList[Series[Sum[Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[Sum[StirlingS1[n, k] (k - 1)! DivisorSigma[1, k], {k, 1, n}], {n, 1, 20}]
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
-
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
A330495
a(n) = Sum_{k=1..n} (-1)^(n-k) * Stirling1(n,k) * (k-1)! * sigma_2(k), where sigma_2 = A001157.
Original entry on oeis.org
1, 6, 37, 307, 2858, 32060, 405830, 5777354, 91400200, 1593023040, 30251766840, 622016655816, 13777150847952, 327040289212320, 8280040187137200, 222696435041359824, 6341359225470493440, 190609840724078576256, 6031297367477133540480, 200389374367707186619776
Offset: 1
-
Table[Sum[(-1)^(n-k) * StirlingS1[n, k] * (k-1)! * DivisorSigma[2, 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)^2), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]
-
a(n) = sum(k=1, n, (-1)^(n-k)*stirling(n, k, 1)*(k-1)!*sigma(k, 2)); \\ Michel Marcus, Dec 16 2019
A330499
Expansion of e.g.f. Sum_{k>=1} log(1 + log(1/(1 - x))^k).
Original entry on oeis.org
0, 1, 2, 13, 71, 558, 5344, 60926, 766898, 10759096, 168848256, 2947203048, 56368708824, 1165246323408, 25802649445728, 609940593443952, 15377212949988624, 412827548455415040, 11764577341464710016, 354392697960438122880, 11237993013428254071936
Offset: 0
-
nmax = 20; CoefficientList[Series[Sum[Log[1+Log[1/(1-x)]^k], {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
Showing 1-4 of 4 results.
