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 A052618 #43 Jan 22 2023 02:36:35
%S A052618 1,2,8,36,216,1440,11520,100800,1008000,10886400,130636800,1676505600,
%T A052618 23471078400,348713164800,5579410636800,94152554496000,
%U A052618 1694745980928000,32011868528640000,640237370572800000,13380961044971520000,294381142989373440000
%N A052618 Expansion of e.g.f. 1/((1-x)^2*(1-x^2)).
%C A052618 Permanent of the n X n (0, 1)-matrix with (i, j)-th entry equal to 0 iff (i=1, j=n), (i=2, j=1), (i=3, j=n), (i=4, j=1), ... - _Simone Severini_, Oct 17 2004
%C A052618 a(n) is the number of runs of odd entries in all permutations of {1,2,...,n+1}. Example: a(2)=8 because in the permutations 123, 132, 213, 231, 312 and 321 we have a total of 2+1+1+1+1+2 runs of odd entries. - _Emeric Deutsch_, Dec 14 2008
%C A052618 a(n) is the number of permutations of [n+2] whose first place is even and last place is odd (or any equivalent definition with two separate places in a permutation). - _Olivier Gérard_, Nov 07 2011
%H A052618 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=563">Encyclopedia of Combinatorial Structures 563</a>.
%F A052618 E.g.f.: -1/(-1+x)^2/(-1+x^2).
%F A052618 Recurrence: {a(0)=1, a(1)=2, (-n^2-5*n-4)*a(n)+a(n+2)-2*a(n+1)=0.}.
%F A052618 a(n) = (1/8*(-1)^(-n)+1/4*n^2+n+7/8)*n! for n>0.
%F A052618 From _Emeric Deutsch_, Dec 14 2008: (Start)
%F A052618 a(n) = n!*floor((n+2)/2)*ceiling((n+2)/2).
%F A052618 a(n) = Sum_{k>=1} (k*A152666(n+1,k)). (End)
%F A052618 a(n) = n!*A002620(n+2). - _R. J. Mathar_, Nov 27 2011
%F A052618 Sum_{n>=0} 1/a(n) = 4*(sinh(1) + gamma - CoshIntegral(1)) - 2 = 4*(A073742 + A001620 - A099284) - 2. - _Amiram Eldar_, Jan 22 2023
%e A052618 The a(2) = 8 permutations of [4] starting with an even number and ending with an odd number are: 2143, 2341, 2413, 2431, 4123, 4213, 4231, 4321.
%p A052618 spec := [S,{S=Prod(Sequence(Z),Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052618 a := proc (n) options operator, arrow: factorial(n)*floor((1/2)*n+1)*ceil((1/2)*n+1) end proc; seq(a(n), n = 0 .. 20); # _Emeric Deutsch_, Dec 14 2008
%t A052618 With[{nn=20},CoefficientList[Series[1/((1-x)^2*(1-x^2)),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Jun 01 2019 *)
%Y A052618 Cf. A002620, A152666.
%Y A052618 Cf. A001620, A073742, A099284.
%K A052618 easy,nonn
%O A052618 0,2
%A A052618 encyclopedia(AT)pommard.inria.fr, Jan 25 2000