A054091 Row sums of A054090.

1, 2, 4, 10, 32, 130, 652, 3914, 27400, 219202, 1972820, 19728202, 217010224, 2604122690, 33853594972, 473950329610, 7109254944152, 113748079106434, 1933717344809380, 34806912206568842, 661331331924808000, 13226626638496160002, 277759159408419360044

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

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

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

Row sums of triangle A208058. - Gary W. Adamson, Feb 22 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Shuzhen Lv and Philip B. Zhang, Joint equidistributions of mesh patterns 123 and 321 with symmetric and antipodal shadings, arXiv:2501.00357 [math.CO], 2024. See p. 15.

Crossrefs

Cf. A000522, A054090, A094816, A208058.

Programs

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

Maple

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
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, 2+(n-1)*a(n-1))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Jun 23 2022

Mathematica

Table[If[n==0, 1, 2*(n-1)!*Sum[1/j!, {j,0,n-1}]], {n,0,30}] (* G. C. Greubel, Jun 23 2022 *)

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 */

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 */

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

Formula

a(n+1) = 2*A000522(n).

a(n+1) = Sum(2*n!/j!, j=0..n). - Zerinvary Lajos, Oct 20 2006

a(n) = 2*floor(e*(n-1)!), n>1. - Gary Detlefs, May 20 2010

a(n) = n*a(n-1) - (n-2)*a(n-2), a(0)=1, a(1)=2. - Vincenzo Librandi, Feb 23 2012