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

%I A166357 #12 Jul 19 2019 14:27:21
%S A166357 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,
%T A166357 1008,0,280,0,42,0,1,5760,0,4032,0,560,0,56,0,1,0,51840,0,12096,0,
%U A166357 1008,0,72,0,1,403200,0,259200,0,30240,0,1680,0,90,0,1
%N A166357 Exponential Riordan array [1+x*arctanh(x), x].
%C A166357 Row sums are A166358. Diagonal sums are A166359.
%H A166357 Andrew Howroyd, <a href="/A166357/b166357.txt">Table of n, a(n) for n = 0..1274</a> (rows 0..49)
%F A166357 Number triangle T(n,k) = [k<=n]*A166356((n-k)/2)*C(n,k)*(1+(-1)^(n-k))/2.
%e A166357 Triangle begins
%e A166357        1;
%e A166357        0,     1;
%e A166357        2,     0,      1;
%e A166357        0,     6,      0,     1;
%e A166357        8,     0,     12,     0,     1;
%e A166357        0,    40,      0,    20,     0,    1;
%e A166357      144,     0,    120,     0,    30,    0,    1;
%e A166357        0,  1008,      0,   280,     0,   42,    0,  1;
%e A166357     5760,     0,   4032,     0,   560,    0,   56,  0,  1;
%e A166357        0, 51840,      0, 12096,     0, 1008,    0, 72,  0, 1;
%e A166357   403200,     0, 259200,     0, 30240,    0, 1680,  0, 90, 0, 1;
%t A166357 (* The function RiordanArray is defined in A256893. *)
%t A166357 RiordanArray[1 + # ArcTanh[#]&, #&, 11, True] // Flatten (* _Jean-François Alcover_, Jul 19 2019 *)
%o A166357 (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
%o A166357 (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
%K A166357 easy,nonn,tabl
%O A166357 0,4
%A A166357 _Paul Barry_, Oct 12 2009