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A054091 Row sums of A054090.

%I A054091 #29 Jan 08 2025 12:53:06
%S A054091 1,2,4,10,32,130,652,3914,27400,219202,1972820,19728202,217010224,
%T A054091 2604122690,33853594972,473950329610,7109254944152,113748079106434,
%U A054091 1933717344809380,34806912206568842,661331331924808000,13226626638496160002,277759159408419360044
%N A054091 Row sums of A054090.
%C A054091 Row sums of A094816 as a triangular array as follows: {1}, {1,1}, {1,3}, {1,1,8}, {6,1,1,24}, {29,10,1,1,89}, ... - _Michael Somos_, Nov 19 2006
%C A054091 a(n) = (n-1)a(n-1)+2, n>0; 2=0*1+2, 4=1*2+2, 10=2*4+2, ... - _Gary Detlefs_, May 20 2010
%C A054091 Row sums of triangle A208058. - _Gary W. Adamson_, Feb 22 2012
%H A054091 Vincenzo Librandi, <a href="/A054091/b054091.txt">Table of n, a(n) for n = 0..100</a>
%H A054091 Shuzhen Lv and Philip B. Zhang, <a href="https://arxiv.org/abs/2501.00357">Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings</a>, arXiv:2501.00357 [math.CO], 2024. See p. 15.
%F A054091 a(n+1) = 2*A000522(n).
%F A054091 a(n+1) = Sum(2*n!/j!, j=0..n). - _Zerinvary Lajos_, Oct 20 2006
%F A054091 a(n) = 2*floor(e*(n-1)!), n>1. - _Gary Detlefs_, May 20 2010
%F A054091 a(n) = n*a(n-1) - (n-2)*a(n-2), a(0)=1, a(1)=2. - _Vincenzo Librandi_, Feb 23 2012
%p A054091 a:= n-> `if`( n=0, 1, add(2*(n-1)!/j!, j=0..n-1)): seq(a(n), n=0..18); # _Zerinvary Lajos_, Oct 20 2006
%p A054091 # second Maple program:
%p A054091 a:= proc(n) option remember;
%p A054091       `if`(n=0, 1, 2+(n-1)*a(n-1))
%p A054091     end:
%p A054091 seq(a(n), n=0..23);  # _Alois P. Heinz_, Jun 23 2022
%t A054091 Table[If[n==0, 1, 2*(n-1)!*Sum[1/j!, {j,0,n-1}]], {n,0,30}] (* _G. C. Greubel_, Jun 23 2022 *)
%o A054091 (PARI) {a(n)= local(A); if(n<1, n==0, A=vector(n); A[1]=2; for(k=1, n-1, A[k+1]=k*A[k]+2); A[n])} /* _Michael Somos_, Nov 19 2006 */
%o A054091 (PARI) {a(n)= if(n<1, n==0, n--; n!*polcoeff( 2*exp(x+x*O(x^n))/(1-x), n))} /* _Michael Somos_, Nov 19 2006 */
%o A054091 (Magma) [n le 2 select n else (n-1)*Self(n-1) -(n-3)*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jun 23 2022
%o A054091 (SageMath) [1]+[2*factorial(n-1)*sum(1/factorial(j) for j in (0..n-1)) for n in (1..30)] # _G. C. Greubel_, Jun 23 2022
%Y A054091 Cf. A000522, A054090, A094816, A208058.
%K A054091 nonn
%O A054091 0,2
%A A054091 _Clark Kimberling_