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