A308346 - OEIS

%I #17 Sep 08 2022 08:46:21

%S 1,0,-2,-6,-10,20,352,2772,18132,104400,469608,238920,-35811048,

%T -730972944,-11436661728,-164609993520,-2294024595312,-31488879303552,

%U -426338226719904,-5626751283423072,-70000948158061728,-745703905072996800,-4142683990211677440,110386551348875714880

%N Expansion of e.g.f. 1/(1 - x)^log(1 - x).

%H Robert Israel, <a href="/A308346/b308346.txt">Table of n, a(n) for n = 0..453</a>

%F a(n) = Sum_{k=0..n} |Stirling1(n,k)|*A067994(k).

%p E:= 1/(1-x)^log(1-x):

%p S:= series(E,x,31):

%p seq(coeff(S,x,j)*j!,j=0..30); # _Robert Israel_, May 22 2019

%t nmax = 23; CoefficientList[Series[1/(1 - x)^Log[1 - x], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Abs[StirlingS1[n, k]] HermiteH[k, 0], {k, 0, n}], {n, 0, 23}]

%t 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}]

%o (PARI) a(n) = sum(k=0, n, abs(stirling(n, k, 1))*polhermite(k, 0)); \\ _Michel Marcus_, May 21 2019

%o (Magma) m:=30; R<x>:=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

%o (Sage) 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

%Y Cf. A000254, A008405, A009199, A052517, A067994, A086660, A087761.

%K sign,look

%O 0,3

%A _Ilya Gutkovskiy_, May 21 2019