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A277604
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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.
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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, 17, 73, 81, 61, 1, 1, 1, 13, 21, 121, 169, 301, 125, 1, 1, 1, 15, 25, 181, 289, 841, 729, 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
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OFFSET
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0,9
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COMMENTS
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It will be interesting using the formulae for k < 0 (attention: signed terms!). Especially for k = -1 see A157674.
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
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LINKS
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FORMULA
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A(k,x) = (x + sqrt(1 - 4*k*x^2))/(1 - (4*k+1)*x^2) for k >= 0.
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.
A(k,x) = 1/(1 - x - 2*k*x^2*C(k*x^2)), k >= 0, where C is the g.f. of A000108.
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.
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
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EXAMPLE
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The terms define the array T(k,n) for k >= 0 and n >= 0, i.e.,
k\n 0 1 2 3 4 5 6 7 8 9 . . .
0: 1 1 1 1 1 1 1 1 1 1 . . .
1: 1 1 3 5 13 25 61 125 295 625 . . .
2: 1 1 5 9 37 81 301 729 2549 6561 . . .
3: 1 1 7 13 73 169 841 2197 10123 28561 . . .
4: 1 1 9 17 121 289 1801 4913 28057 83521 . . .
5: 1 1 11 21 181 441 3301 9261 63071 194481 . . .
6: 1 1 13 25 253 625 5461 15625 123565 390625 . . .
7: 1 1 15 29 337 841 8401 24389 219619 707281 . . .
8: 1 1 17 33 433 1089 12241 35937 362993 1185921 . . .
9: 1 1 19 37 541 1369 17101 50653 567127 1874161 . . .
etc.
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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