This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A120765 #39 Apr 21 2025 13:24:25
%S A120765 0,1,0,5,24,209,2120,25829,365456,5895105,106794992,2147006949,
%T A120765 47436635752,1142570789073,29797622256376,836527783016197,
%U A120765 25153234375160992,806519154686509057,27470342073410272608,990496662138073867333,37692249497898323450424
%N A120765 Expansion of e.g.f. -exp(-x)*log(1-2*x)/2.
%C A120765 Inverse binomial transform of even double factorials (A000165) with 0 prepended: [0, 1, 2, 8, 48 ...].
%H A120765 Alois P. Heinz, <a href="/A120765/b120765.txt">Table of n, a(n) for n = 0..300</a>
%H A120765 Roland Bacher, <a href="https://doi.org/10.37236/2522">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
%F A120765 E.g.f.: -e^(-x)*log(1-2*x)/2.
%F A120765 a(n) = Sum_{i=0..n-1} (-1)^(n-1-i) * C(n,i+1) * i! * 2^i.
%F A120765 a(n) ~ (n-1)! * 2^(n-1) * exp(-1/2). - _Vaclav Kotesovec_, Oct 08 2013
%F A120765 From _Vladimir Reshetnikov_, Oct 28 2015: (Start)
%F A120765 a(n) = Sum_{k=0..n-1} A000354(k)*(-1)^(n+k+1).
%F A120765 Recurrence: a(0) = 0, a(1) = 1, a(2) = 0, a(n) = 2*(n-2)*a(n-3) + (4*n-7)*a(n-2) + 2*(n-2)*a(n-1). (End)
%F A120765 a(n) = (-1)^(n+1)*n*hypergeom([1-n,1,1],[2],2). - _Peter Luschny_, May 09 2017
%p A120765 a:= proc(n) option remember; `if`(n<3, n*(2-n),
%p A120765 (2*n-4)*(a(n-1) +a(n-3)) +(4*n-7)*a(n-2))
%p A120765 end:
%p A120765 seq(a(n), n=0..25); # _Alois P. Heinz_, Oct 08 2013
%p A120765 A120765 := n -> (-1)^(n+1)*n*hypergeom([1-n,1,1],[2],2):
%p A120765 seq(simplify(A120765(n)), n=0..20); # _Peter Luschny_, May 09 2017
%t A120765 CoefficientList[Series[-E^(-x)*Log[1-2*x]/2, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 08 2013 *)
%o A120765 (PARI) my(x='x+O('x^33)); concat([0],Vec(serlaplace(-exp(-x)*log(1-2*x)/2))) \\ _Joerg Arndt_, Jun 29 2015
%o A120765 (PARI) vector(30, n, n--; sum(k=0, n-1, (-1)^(n-1-k) * binomial(n,k+1) * k! * 2^k)) \\ _Altug Alkan_, Oct 28 2015
%Y A120765 Cf. A000165, A000354.
%K A120765 nonn
%O A120765 0,4
%A A120765 _Max Alekseyev_, Jul 03 2006
%E A120765 Formula corrected by _Max Alekseyev_, Sep 15 2009