A166357 Exponential Riordan array [1+x*arctanh(x), x].

1, 0, 1, 2, 0, 1, 0, 6, 0, 1, 8, 0, 12, 0, 1, 0, 40, 0, 20, 0, 1, 144, 0, 120, 0, 30, 0, 1, 0, 1008, 0, 280, 0, 42, 0, 1, 5760, 0, 4032, 0, 560, 0, 56, 0, 1, 0, 51840, 0, 12096, 0, 1008, 0, 72, 0, 1, 403200, 0, 259200, 0, 30240, 0, 1680, 0, 90, 0, 1

Offset: 0

Author

Paul Barry, Oct 12 2009

Keywords

• easy
• nonn
• tabl

Comments

Row sums are A166358. Diagonal sums are A166359.

Examples

			Triangle begins
       1;
       0,     1;
       2,     0,      1;
       0,     6,      0,     1;
       8,     0,     12,     0,     1;
       0,    40,      0,    20,     0,    1;
     144,     0,    120,     0,    30,    0,    1;
       0,  1008,      0,   280,     0,   42,    0,  1;
    5760,     0,   4032,     0,   560,    0,   56,  0,  1;
       0, 51840,      0, 12096,     0, 1008,    0, 72,  0, 1;
  403200,     0, 259200,     0, 30240,    0, 1680,  0, 90, 0, 1;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274 (rows 0..49)

Programs

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1 + # ArcTanh[#]&, #&, 11, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

PARI

T(n,k)={binomial(n,k)*(n-k)!*polcoef(1 + x*atanh(x + O(x^max(1, n-k))), n-k)} \\ Andrew Howroyd, Aug 17 2018

PARI

T(n,k)=if(k>=n, n==k, binomial(n, k)*if((n-k)%2, 0, (n-k-1)! + (n-k-2)!)) \\ Andrew Howroyd, Aug 17 2018

Formula

Number triangle T(n,k) = [k<=n]*A166356((n-k)/2)*C(n,k)*(1+(-1)^(n-k))/2.