A352069
Expansion of e.g.f. 1 / (1 + log(1 - 3*x) / 3).
Original entry on oeis.org
1, 1, 5, 42, 492, 7374, 134478, 2887128, 71281656, 1988802720, 61860849552, 2121993490176, 79566300371952, 3237181141173264, 142019158472311248, 6682603650677875584, 335698708873243355136, 17930674324049810882688, 1014685181110897126616448, 60641642160287342580586752
Offset: 0
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nmax = 19; CoefficientList[Series[1/(1 + Log[1 - 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! (-3)^(n - k), {k, 0, n}], {n, 0, 19}]
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my(x='x+O('x^25)); Vec(serlaplace(1/(1+log(1-3*x)/3))) \\ Michel Marcus, Mar 02 2022
A352074
a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * (-n)^(n-k).
Original entry on oeis.org
1, 1, 4, 42, 904, 34070, 2019888, 174588120, 20804747136, 3276218158560, 659664288364800, 165425062846302336, 50574549124825998336, 18520126461205806360144, 8003819275469728355033088, 4031020344281171589447408000, 2340375822778055527109749211136
Offset: 0
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Unprotect[Power]; 0^0 = 1; Table[Sum[StirlingS1[n, k] k! (-n)^(n - k), {k, 0, n}], {n, 0, 16}]
Join[{1}, Table[n! SeriesCoefficient[1/(1 + Log[1 - n x]/n), {x, 0, n}], {n, 1, 16}]]
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a(n) = sum(k=0, n, stirling(n, k, 1)*k!*(-n)^(n-k)); \\ Michel Marcus, Mar 02 2022
A354751
Expansion of e.g.f. 1 / (1 - log(1 + 4*x) / 4).
Original entry on oeis.org
1, 1, -2, 14, -152, 2264, -42832, 982512, -26484096, 820207488, -28692711168, 1118821622016, -48112717347840, 2261868010650624, -115400220781209600, 6350152838136428544, -374874781697133871104, 23632196147497381625856, -1584445791263626895228928
Offset: 0
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nmax = 18; CoefficientList[Series[1/(1 - Log[1 + 4 x]/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 4^(n - k), {k, 0, n}], {n, 0, 18}]
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my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+4*x)/4))) \\ Michel Marcus, Jun 06 2022
A367853
Expansion of e.g.f. 1/(1 - x + log(1 - 4*x)/4).
Original entry on oeis.org
1, 2, 12, 128, 1952, 38464, 926336, 26323968, 861419520, 31882358784, 1316275003392, 59954841649152, 2985997926727680, 161401148097036288, 9408988894966579200, 588381964243109412864, 39285329204482179858432, 2789234068575581984784384
Offset: 0
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 4^(j-1)*(j-1)!*binomial(i, j)*v[i-j+1])); v;
Showing 1-4 of 4 results.