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A300490 Expansion of e.g.f. -exp(-x)*log(1 - x)/(1 - x).

%I A300490 #13 Mar 30 2022 12:05:59
%S A300490 0,1,1,5,20,109,689,5053,42048,391641,4036697,45618341,560889988,
%T A300490 7454314789,106488455033,1627269878557,26487441519584,457532446622001,
%U A300490 8359188190686609,161056273132588933,3263644496701880404,69389030027882288861,1544501472271318499105
%N A300490 Expansion of e.g.f. -exp(-x)*log(1 - x)/(1 - x).
%C A300490 Inverse binomial transform A000254.
%H A300490 Alois P. Heinz, <a href="/A300490/b300490.txt">Table of n, a(n) for n = 0..450</a>
%H A300490 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A300490 a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*k!*H(k), where H(k) is the k-th harmonic number.
%F A300490 a(n) ~ n! * (log(n) + gamma) / exp(1), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jun 23 2018
%F A300490 Recurrence: a(0) = 0, a(1) = 1, a(2) = 1, a(3) = 5, a(n+4) = (2*n+5)*a(n+3) - (n^2+2*n-1)*a(n+2) - (2*n+3)*(n+2)*a(n+1) - (n+2)*(n+1)*a(n). - _Vladimir Reshetnikov_, Mar 30 2022
%e A300490 -exp(-x)*log(1 - x)/(1 - x) = x/1! + x^2/2! + 5*x^3/3! + 20*x^4/4! + 109*x^5/5! + 689*x^6/6! + 5053*x^7/7! + ...
%p A300490 b:= proc(n) option remember; `if`(n<2, n, n*b(n-1)+(n-1)!) end:
%p A300490 a:= proc(n) add(b(k)*(-1)^(n-k)*binomial(n, k), k=0..n) end:
%p A300490 seq(a(n), n=0..25);  # _Alois P. Heinz_, Mar 07 2018
%t A300490 nmax = 22; CoefficientList[Series[-Exp[-x] Log[1 - x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
%t A300490 Table[Sum[(-1)^(n - k) Binomial[n,k] k! HarmonicNumber[k], {k, 1, n}], {n, 0, 22}]
%t A300490 a[0] = 0; a[1] = a[2] = 1; a[3] = 5; a[n_Integer] := a[n] = (2 n - 3) a[n - 1] - (n^2 - 6 n + 7) a[n - 2] - (n - 2) (2 n - 5) a[n - 3] - (n - 3) (n - 2) a[n - 4]; Table[a[n], {n, 0, 22}] (* _Vladimir Reshetnikov_, Mar 30 2022 *)
%Y A300490 Cf. A000254, A073596.
%K A300490 nonn
%O A300490 0,4
%A A300490 _Ilya Gutkovskiy_, Mar 07 2018