%I #48 Feb 17 2024 04:03:27

%S 2,5,14,54,264,1560,10800,85680,766080,7620480,83462400,997920000,

%T 12933043200,180583603200,2702527027200,43153254144000,

%U 732297646080000,13160434839552000,249692574523392000,4987449116762112000,104614786351595520000,2299092397726924800000

%N Expansion of e.g.f. (2+x-x^2)/(1-x)^2.

%C a(1) is 5 and gives the row number in the table of 0-origin permutations of order 3 in which the first 3 items are reversed. Row 5 of this table is 2 1 0. a(2) is 14 and gives the row number in the table of 0-origin permutations of order 4 in which the first three items are reversed. Row 14 of this table is 2 1 0 3.... a(6) is 10800 and gives the row number in the table of 0-origin permutations of order 8 in which the first 3 items are reversed. Row 10800 of this table is 2 1 0 3 4 5 6 7. Et cetera. - Eugene McDonnell (eemcd(AT)mac.com), Dec 03 2004

%C In factorial base representation (A007623) the terms of this sequence are written as: 10, 21, 210, 2100, 21000, 210000, ... From a(1) = 5 = "21" onward each term begins always with "21", which is then followed by n-1 zeros. - _Antti Karttunen_, Sep 24 2016

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=596">Encyclopedia of Combinatorial Structures 596</a>.

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.

%F a(n) = (3+2*n)*n!.

%F E.g.f.: -(-x+x^2-2)/(-1+x)^2.

%F Recurrence: a(0)=2, a(1)=5, (-7*n-5-2*n^2)*a(n)+(3+2*n)*a(n+1)=0 for n>=1.

%F a(n) = A129326(n), n>1. - _R. J. Mathar_, Jun 14 2008

%F a(n) = (n+1)*a(n-1) - 2*A001048(n-1). - _Gary Detlefs_, Dec 16 2009

%F a(0) = 2; for n >= 1, a(n) = 2*(n+1)! + n! - _Antti Karttunen_, Sep 24 2016

%F From _Amiram Eldar_, Feb 17 2024: (Start)

%F Sum_{n>=0} 1/a(n) = 1/6 + e/2 - erfi(1)*sqrt(Pi)/4, where erfi is the imaginary error function.

%F Sum_{n>=0} (-1)^n/a(n) = 1/6 - 1/(2*e) + erf(1)*sqrt(Pi)/4, where erf is the error function. (End)

%p spec := [S,{S=Prod(Sequence(Z),Union(Z,Sequence(Z),Sequence(Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%t f[n_] := (3 + 2 n) n!; f[0] = 2; Array[f, 19, 0]

%t a[n_] := a[n] = a[n - 1]*n (2 n + 3)/(2 n + 1); a[0] = 2; a[1] = 5; Array[ a, 19, 0] (* _Robert G. Wilson v_ *)

%t With[{nn=20},CoefficientList[Series[(2+x-x^2)/(1-x)^2,{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, Nov 09 2017 *)

%o (PARI) a(n)=if(n<=1,[2,5][n+1], a(n-1)*(n*(2*n+3))/(2*n+1) );

%o for(n=0,11,print1(a(n),", "))

%o (Scheme) (define (A052649 n) (if (zero? n) 2 (+ (A000142 n) (* 2 (A000142 (+ 1 n)))))) ;; _Antti Karttunen_, Sep 24 2016

%Y Cf. A000142, A001048, A007623, A129326.

%Y Row 4 of A276955 (from a(1)=5 onward).

%K easy,nonn

%O 0,1

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000