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%I #18 Dec 07 2019 12:18:28
%S 1,1,1,1,3,1,1,5,5,1,1,7,13,3,1,1,9,25,5,-3,1,1,11,41,7,-59,3,1,1,13,
%T 61,9,-263,5,29,1,1,15,85,11,-759,7,805,3,1,1,17,113,13,-1739,9,6649,
%U 5,-131,1,1,19,145,15,-3443,11,31241,7,-12155,3,1,1,21,181,17,-6159,13,106261,9,-200711,5,765,1
%N Array of coefficients a(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = ((2*k+1)*x+sqrt(1+4*k*(k+1)*x^2))/(1-x^2), k>=0.
%C The A(k,x) satisfy A(k,x)^2 = 1+(4*k+2)*x*A(k,x)+x^2*A(k,x)^2 for k>=0.
%C The terms of odd-numbered columns a(k,2*n+1) are simple with (2*k+1)*x/(1-x^2), analogous the even-numbered columns a(k,2*n) with the o.g.f. of A000108.
%F a(k,0) = 1 and a(k,2*n+2) = 1-2*(Sum_{i=0..n} A000108(i)*(-k*(k+1))^(i+1)) and a(k,2*n+1) = 2*k+1 for k >= 0 and n >= 0.
%F A(k,x) = (1+(2*k+1)*x+2*k*(k+1)*x^2*C(-k*(k+1)*x^2))/(1-x^2) for k >= 0, where C is the o.g.f. of A000108.
%F A(k,x)*A(k,-x) = 1/(1-x^2) for k >= 0.
%F Conjecture: a(k,2*n+2) = 1+2*k+2*(-k)^(n+2)*(Sum_{i=0..n} A234950(n,i)*k^i) for k>=0 and n>=0. - _Werner Schulte_, Aug 03 2017
%e The terms define the array a(k,n) for k >= 0 and n >= 0, i.e.,
%e k\n 0 1 2 3 4 5 6 7 8 9 10 11 ...
%e 0: 1 1 1 1 1 1 1 1 1 1 1 1 ...
%e 1: 1 3 5 3 -3 3 29 3 -131 3 765 3 ...
%e 2: 1 5 13 5 -59 5 805 5 -12155 5 205573 5 ...
%e 3: 1 7 25 7 -263 7 6649 7 -200711 7 6766585 7 ...
%e 4: 1 9 41 9 -759 9 31241 9 -1568759 9 88031241 9 ...
%e 5: 1 11 61 11 -1739 11 106261 11 -7993739 11 672406261 11 ...
%e 6: 1 13 85 13 -3443 13 292909 13 -30824051 13 ...
%e 7: 1 15 113 15 -6159 15 696305 15 -97648655 15 ...
%e 8: 1 17 145 17 -10223 17 1482769 17 -267255791 17 ...
%e 9: 1 19 181 19 -16019 19 2899981 19 ...
%e 10: 1 21 221 21 -23979 21 5300021 21 ...
%e etc.
%e The formal power series corresponding to row 2 is A(2,x) = 1+5*x+13*x^2+5*x^3 ..
%e The terms define the triangle T(k,n) = a(k-n,n) for 0 <= n <=k, i.e.,
%e k\n 0 1 2 3 4 5 ...
%e 0: 1
%e 1: 1 1
%e 2: 1 3 1
%e 3: 1 5 5 1
%e 4: 1 7 13 3 1
%e 5: 1 9 25 5 -3 1
%e etc.
%t A[k_, n_]:=If[n==0, 1, If[EvenQ[n], 1 - 2 Sum[CatalanNumber[i] (-k(k + 1))^(i + 1), {i, 0, (n - 2)/2}], 2k + 1]]; Table[A[n - k, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _Indranil Ghosh_, Aug 03 2017 *)
%o (Python)
%o from sympy import catalan
%o def A(k, n): return 1 if n==0 else 1 - 2*sum([catalan(i)*(-k*(k + 1))**(i + 1) for i in range(n/2)]) if n%2==0 else 2*k + 1
%o for n in range(13): print [A(n - k, k) for k in range(n + 1)] # _Indranil Ghosh_, Aug 03 2017
%Y Cf. A000108, A234950.
%K sign,easy,tabl
%O 0,5
%A _Werner Schulte_, Nov 04 2016
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