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%I #54 Aug 10 2017 03:59:57
%S 1,1,1,1,1,1,1,1,3,1,1,1,5,5,1,1,1,7,9,13,1,1,1,9,13,37,25,1,1,1,11,
%T 17,73,81,61,1,1,1,13,21,121,169,301,125,1,1,1,15,25,181,289,841,729,
%U 295,1,1,1,17,29,253,441,1801,2197,2549,625,1,1,1,19,33,337,625,3301,4913,10123,6561,1447,1
%N Array of coefficients T(k,n) of the formal power series A(k,x) read by upwards antidiagonals, where A(k,x) = sqrt(1 + 2*x*A(k,x) + (4*k+1)*x^2*(A(k,x))^2), k >= 0.
%C For k = 0 see A000012, for k = 1 see A098615, and for k = 2 see A200376.
%C It will be interesting using the formulae for k < 0 (attention: signed terms!). Especially for k = -1 see A157674.
%C If G is the g.f. of central binomial coefficients (see A000984) and B(k,x) = G(k*x^2), then B(k,x) = A(k,x)/(1+x*A(k,x)) and A(k,x) = B(k,x) / (1-x*B(k,x)) for k >= 0. - _Werner Schulte_, Aug 07 2017
%F A(k,x) = (x + sqrt(1 - 4*k*x^2))/(1 - (4*k+1)*x^2) for k >= 0.
%F T(k,0) = 1 and T(k,2*n+2) = (4*k+1)^(n+1)-2*(Sum_{i=0..n} A000108(i)*k^(i+1)* (4*k+1)^(n-i)), and T(k,2*n+1) = (4*k+1)^n for k >= 0 and n >= 0.
%F A(k,x) = 1/(1 - x - 2*k*x^2*C(k*x^2)), k >= 0, where C is the g.f. of A000108.
%F Conjecture: If B(k,n) satisfy B(k,0) = B(k,1) = 1 and B(k,n+2) = B(k,n+1) + k*B(k,n) for k >= 0 and n >= 0 (generalized Fibonacci numbers, see A015441) and G(k,x) = Sum_{n>=0} A000108(n)*B(k,n)*x^n for k >= 0, then you will have (1): A(k,x*G(k,x)) = G(k,x) and (2): G(k,x/A(k,x)) = A(k,x) for k >= 0. Especially for k = 1 see A098615 and for k = 2 see A200376.
%F Conjecture: T(k,2*n) = Sum_{i=0..n} A046521(n,i)*k^(n-i) for k, n >= 0. - _Werner Schulte_, Aug 02 2017
%F Recurrence: T(k,2*n+2) = (4*k+1)*T(k,2*n)-2*k^(n+1)*A000108(n) with initial value T(k,0) = 1 for k >= 0 and n >= 0. - _Werner Schulte_, Aug 09 2017
%F T(k,n) = Sum_{i=0..n} A111959(n,i)*k^((n-i)/2) for k >= 0 and n >= 0. - _Werner Schulte_, Aug 09 2017
%e The terms define the array T(k,n) for k >= 0 and n >= 0, i.e.,
%e k\n 0 1 2 3 4 5 6 7 8 9 . . .
%e 0: 1 1 1 1 1 1 1 1 1 1 . . .
%e 1: 1 1 3 5 13 25 61 125 295 625 . . .
%e 2: 1 1 5 9 37 81 301 729 2549 6561 . . .
%e 3: 1 1 7 13 73 169 841 2197 10123 28561 . . .
%e 4: 1 1 9 17 121 289 1801 4913 28057 83521 . . .
%e 5: 1 1 11 21 181 441 3301 9261 63071 194481 . . .
%e 6: 1 1 13 25 253 625 5461 15625 123565 390625 . . .
%e 7: 1 1 15 29 337 841 8401 24389 219619 707281 . . .
%e 8: 1 1 17 33 433 1089 12241 35937 362993 1185921 . . .
%e 9: 1 1 19 37 541 1369 17101 50653 567127 1874161 . . .
%e etc.
%Y Cf. A000012, A000045, A000108, A000984, A015441, A046521, A098615, A111959, A200376.
%K nonn,easy,tabl
%O 0,9
%A _Werner Schulte_, Oct 29 2016
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