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E.g.f.: log(1 + log(1 - x))^2 / 2.
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1, 6, 40, 315, 2908, 30989, 375611, 5112570, 77305024, 1286640410, 23387713930, 461187042992, 9808283703684, 223833267479764, 5456669750439788, 141540592345674800, 3892707724320135616, 113153294901088030320, 3466501398608272647984, 111636571036702743967104, 3770483138507706753943584
FORMULA
a(n) = Sum_{k=2..n} |Stirling1(n, k) * Stirling1(k, 2)|.
a(n) = Sum_{k=2..n} |Stirling1(n, k)| * (k-1)! * H(k-1), where H(k) is the k-th harmonic number.
a(n) ~ sqrt(2*Pi) * log(n) * n^(n - 1/2) / (exp(1) - 1)^n * (1 + (gamma - log(exp(1) - 1))/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 15 2021
MATHEMATICA
nmax = 22; CoefficientList[Series[Log[1 + Log[1 - x]]^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
Table[Sum[Abs[StirlingS1[n, k] StirlingS1[k, 2]], {k, 2, n}], {n, 2, 22}]
Expansion of e.g.f. -log(1 + log(1 - x))/(1 + log(1 - x)).
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0, 1, 4, 22, 155, 1333, 13541, 158688, 2107682, 31291894, 513590170, 9234669420, 180534475832, 3812852144788, 86517295628188, 2099170738243328, 54233876338638192, 1486517654443664016, 43084555863325589232, 1316588795487600071904, 42306543064537291007424, 1426115146736949130634400
FORMULA
a(n) = Sum_{k=1..n} |Stirling1(n,k)|*H(k)*k!, where H(k) is the k-th harmonic number.
a(n) ~ sqrt(2*Pi) * log(n) * n^(n + 1/2) / (exp(1)-1)^(n+1). - Vaclav Kotesovec, Jun 23 2018
EXAMPLE
E.g.f.: A(x) = x + 4*x^2/2! + 22*x^3/3! + 155*x^4/4! + 1333*x^5/5! + 13541*x^6/6! + ...
MAPLE
H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*H(k)*k!, k=1..n):
MATHEMATICA
nmax = 21; CoefficientList[Series[-Log[1 + Log[1 - x]]/(1 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HarmonicNumber[k] k!, {k, 0, n}], {n, 0, 21}]
E.g.f.: log(1 - log(1 - x))^2 / 2.
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1
1, 0, 4, 5, 58, 217, 2035, 13470, 134164, 1243770, 14129410, 164244808, 2151576620, 29671566836, 444758323628, 7055358559376, 119546765395744, 2139179551573104, 40486788832168944, 805969129348431936, 16860672502118423136, 369459637224850523808, 8467140450141232328160
FORMULA
a(n) = Sum_{k=2..n} |Stirling1(n, k)| * Stirling1(k, 2).
a(n) = (-1)^n * Sum_{k=2..n} Stirling1(n, k) * (k-1)! * H(k-1), where H(k) is the k-th harmonic number.
MATHEMATICA
nmax = 24; CoefficientList[Series[Log[1 - Log[1 - x]]^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
Table[Sum[Abs[StirlingS1[n, k]] StirlingS1[k, 2], {k, 2, n}], {n, 2, 24}]
a(n) = n! * Sum_{k=1..n} Stirling1(n,k) * H(k), where H(k) is the k-th harmonic number.
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0, 1, 1, -4, 38, -646, 17124, -651120, 33563760, -2251415376, 190506294720, -19843054116480, 2494435702953600, -372324067662349440, 65089674982557308160, -13172994619821785548800, 3055455516855073351219200, -805168341051328705189939200
FORMULA
Sum_{n>=0} a(n) * x^n / n!^2 = Sum_{n>=1} H(n) * log(1+x)^n / n!.
MATHEMATICA
Table[n! Sum[StirlingS1[n, k] HarmonicNumber[k], {k, 1, n}], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Sum[HarmonicNumber[k] Log[1 + x]^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
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