A211205
n-th derivative of x^(x^(x^(x^(x^x)))) at x=1.
Original entry on oeis.org
1, 1, 2, 9, 56, 480, 5094, 60494, 823528, 12365424, 206078880, 3745686912, 74083090872, 1579529362944, 36165466533000, 884104045301640, 22992315801392064, 633547543117707648, 18439576158792912192, 565162707747635408448, 18194047307015185486080
Offset: 0
Cf.
A005727,
A008405,
A176118,
A179230,
A179405,
A179505,
A215522,
A215524,
A215629,
A215643,
A215691,
A215704,
A215705,
A215706,
A215707,
A215708,
A215709,
A215710,
A215522,
A295106.
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a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^(x^(x^x))))), x, n+1), x, n):
seq(a(n), n=0..20);
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NestList[ Factor[ D[#1, x]] &, x^x^x^x^x^x, 9] /. (x -> 1) (* or quicker *)
Range[0, 20]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x), {x, 0, 20}], x]
A300491
Expansion of e.g.f. log(1 - log(1 - x)/(1 - x)).
Original entry on oeis.org
0, 1, 2, 4, 9, 28, 140, 936, 6902, 54160, 467784, 4578000, 50434032, 609309504, 7921524624, 110242136928, 1643101763760, 26192405980416, 444523225673472, 7989603260143104, 151483589818925184, 3022296286833907200, 63326051483436129024, 1390571693776506751488
Offset: 0
log(1 - log(1 - x)/(1 - x)) = x/1! + 2*x^2/2! + 4*x^3/3! + 9*x^4/4! + 28*x^5/5! + 140*x^6/6! + 936*x^7/7! + ...
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b:= proc(n) option remember; `if`(n<2, n, n*b(n-1)+(n-1)!) end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-add(
a(j)*j*binomial(n, j)*b(n-j), j=1..n-1)/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Mar 07 2018
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nmax = 23; CoefficientList[Series[Log[1 - Log[1 - x]/(1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = n! HarmonicNumber[n] - Sum[k Binomial[n, k] (n - k)! HarmonicNumber[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 23}]
A308346
Expansion of e.g.f. 1/(1 - x)^log(1 - x).
Original entry on oeis.org
1, 0, -2, -6, -10, 20, 352, 2772, 18132, 104400, 469608, 238920, -35811048, -730972944, -11436661728, -164609993520, -2294024595312, -31488879303552, -426338226719904, -5626751283423072, -70000948158061728, -745703905072996800, -4142683990211677440, 110386551348875714880
Offset: 0
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m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)^Log(1/(1-x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 21 2019
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E:= 1/(1-x)^log(1-x):
S:= series(E,x,31):
seq(coeff(S,x,j)*j!,j=0..30); # Robert Israel, May 22 2019
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nmax = 23; CoefficientList[Series[1/(1 - x)^Log[1 - x], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] HermiteH[k, 0], {k, 0, n}], {n, 0, 23}]
a[n_] := a[n] = -2 Sum[(k - 1)! HarmonicNumber[k - 1] Binomial[n - 1, k - 1] a[n - k], {k, 2, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
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a(n) = sum(k=0, n, abs(stirling(n, k, 1))*polhermite(k, 0)); \\ Michel Marcus, May 21 2019
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m = 30; T = taylor((1-x)^log(1/(1-x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 21 2019
A308535
Expansion of e.g.f. 1/(1 - x)^log(1 + x) (even powers only).
Original entry on oeis.org
1, 2, 22, 608, 31764, 2695992, 338441112, 58961602464, 13614906576528, 4024831155397536, 1482492491866434912, 665729215100873644800, 358022910151079384324928, 227174478580352888344068480, 167941710127005880795828894080, 143087068385495604780364250426880
Offset: 0
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nmax = 15; Table[(CoefficientList[Series[1/(1 - x)^Log[1 + x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A347978
E.g.f.: 1/(1 + x)^(1/(1 - x)).
Original entry on oeis.org
1, -1, 0, -3, 4, -30, 186, -630, 11600, -26712, 1005480, -2581920, 117196872, -485308824, 17734457664, -131070696120, 3387342915840, -43890398953920, 801577841697216, -17363169328243392, 233460174245351040, -7968629225100337920, 84363134551361043840
Offset: 0
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nmax = 22; CoefficientList[Series[1/(1 + x)^(1/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
A024167[n_] := n! Sum[(-1)^(k + 1)/k, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 1, k - 1] A024167[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
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my(x='x+O('x^30)); Vec(serlaplace(1/(1+x)^(1/(1-x)))) \\ Michel Marcus, Sep 22 2021
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