A308346 Expansion of e.g.f. 1/(1 - x)^log(1 - x).

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

Author

Ilya Gutkovskiy, May 21 2019

Keywords

• sign
• look

Links

• Robert Israel, Table of n, a(n) for n = 0..453

Crossrefs

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

Programs

Magma

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

Maple

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

Mathematica

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}]

PARI

a(n) = sum(k=0, n, abs(stirling(n, k, 1))*polhermite(k, 0)); \\ Michel Marcus, May 21 2019

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

Formula

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