cp's OEIS Frontend

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.

Showing 1-2 of 2 results.

A054091 Row sums of A054090.

Original entry on oeis.org

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

Views

Author

Clark Kimberling

Keywords

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

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

A114219 Number triangle T(n,k) = (k-(k-1)*0^(n-k))*[k<=n].

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 3, 1, 0, 1, 2, 3, 4, 1, 0, 1, 2, 3, 4, 5, 1, 0, 1, 2, 3, 4, 5, 6, 1, 0, 1, 2, 3, 4, 5, 6, 7, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1
Offset: 0

Views

Author

Paul Barry, Nov 18 2005

Keywords

Comments

Row sums are n*(n-1)/2+1 (essentially A000124). Diagonal sums are A114220. First difference triangle of A077028, when this is viewed as a number triangle.
From R. J. Mathar, Mar 22 2013: (Start)
The matrix inverse is
1;
0, 1;
0, -1, 1;
0, 1, -2, 1;
0, -2, 4, -3, 1;
0, 6, -12, 9, -4, 1;
0, -24, 48, -36, 16, -5, 1;
0, 120, -240, 180, -80, 25, -6, 1;
0, -720, 1440, -1080, 480, -150, 36, -7, 1;
... apparently related to A208058. (End)
Number of permutations of length n avoiding simultaneously the patterns 132 and 321 with k left-to-right maxima (resp., right-to-left minima). A left-to-right maximum (resp., right-to-left minimum) in a permutation p(1)p(2)...p(n) is a position i such that p(j) < p(i) for all j < i (resp., p(j) > p(i) for all j > i). - Sergey Kitaev, Nov 18 2023

Examples

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 1, 2, 1;
  0, 1, 2, 3, 1;
  0, 1, 2, 3, 4, 1;
  0, 1, 2, 3, 4, 5, 1;
  0, 1, 2, 3, 4, 5, 6, 1;
  0, 1, 2, 3, 4, 5, 6, 7, 1;
  ...

Links

Crossrefs

Cf. A000124, A114220, A077028.

Programs

  • Maple
    A114219 := proc(n,k)
    if k < 0 or k > n then
        0;
    elif n = k then
        1;
    else
        k ;
    end if;
end proc: # R. J. Mathar, Mar 22 2013

Formula

G.f.: (1-x-u*x + 2u*x^2)/((1-x)(1-u*x)^2), where x records length and u records left-to-right maxima (or right-to-left minima). - Sergey Kitaev, Nov 18 2023
Showing 1-2 of 2 results.

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