A304494
Expansion of e.g.f. Product_{k>=1} (1 + x^k)^H(k), where H(k) is the k-th harmonic number.
Original entry on oeis.org
1, 1, 3, 20, 103, 899, 8143, 84678, 975049, 13082993, 186340631, 2878977408, 48899305783, 876721463435, 16971889682707, 349059348881834, 7565120836998801, 173313418321443809, 4197655086606145387, 106097089652021765356, 2816940203630838490791, 78147038018470085005235
Offset: 0
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nmax = 21; CoefficientList[Series[Product[(1 + x^k)^HarmonicNumber[k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d HarmonicNumber[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
A305201
Expansion of e.g.f. Product_{k>=1} 1/(1 - H(k)*x^k), where H(k) is the k-th harmonic number.
Original entry on oeis.org
1, 1, 5, 26, 208, 1644, 18728, 201466, 2809672, 39505800, 647509992, 10851033984, 210456343392, 4090234000800, 89123794754304, 2000019423403824, 48674645933985408, 1217362548455301504, 32913123947574009984, 910006995701419453440, 26898048642355515339264
Offset: 0
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nmax = 20; CoefficientList[Series[Product[1/(1 - HarmonicNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[HarmonicNumber[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d HarmonicNumber[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 20}]
A304496
Expansion of e.g.f. Product_{k>=1} (1 - x^k)^H(k), where H(k) is the k-th harmonic number.
Original entry on oeis.org
1, -1, -3, -2, 3, 261, 745, 12412, 16289, -260081, -5424199, -96985734, -2047127621, -17402659299, -84365982987, -2937186832544, 39650368238977, 1047895936025183, 35975009604881845, 638531451763185398, 14668256344792565331, 248159858571597211093, 6320237684944085611809
Offset: 0
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nmax = 22; CoefficientList[Series[Product[(1 - x^k)^HarmonicNumber[k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[-Sum[d HarmonicNumber[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
A305128
Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^AH(k), where AH(k) is the k-th alternating harmonic number.
Original entry on oeis.org
1, 1, 3, 14, 79, 539, 4663, 42468, 457945, 5433281, 71036231, 994289658, 15544425103, 253283689619, 4489180389835, 84521336758904, 1687130833152561, 35365641206048129, 790065486354237643, 18340253632236738022, 449655289227002010351, 11492300073384698090795, 306803167368168113022271
Offset: 0
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nmax = 22; CoefficientList[Series[Product[1/(1 - x^k)^Sum[(-1)^(j + 1)/j, {j, 1, k}], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d ((-1)^(d + 1) LerchPhi[-1, 1, d + 1] + Log[2]), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
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