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A307810
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Expansion of 1/AGM(1-64*x, sqrt((1-16*x)*(1-256*x))).
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2
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1, 100, 13924, 2371600, 453093796, 92598490000, 19745403216400, 4333667896360000, 971177275449892900, 221106619001508490000, 50967394891692703241104, 11866732390447357481358400, 2785834789480617203561744656, 658549235163074008904405646400
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OFFSET
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0,2
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COMMENTS
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Also the squares of coefficients in g.f. 1/sqrt((1-4*x)*(1-16*x)).
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LINKS
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FORMULA
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a(n) = A307695(n)^2 = (Sum_{k=0..n} 4^(n-k)*3^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} 16^(n-k)*(-3)^k*binomial(n,k)*binomial(2k,k))^2.
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MATHEMATICA
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a[n_] := Sum[4^(n-k) * 3^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 14, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
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PROG
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(PARI) N=20; x='x+O('x^N); Vec(1/agm(1-64*x, sqrt((1-16*x)*(1-256*x))))
(PARI) {a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))^2}
(PARI) {a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))^2}
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CROSSREFS
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(Sum_{k=0..n} c^(n-k)*e^k*binomial(n,k)*binomial(2k,k))^2 = (Sum_{k=0..n} d^(n-k)*(-e)^k*binomial(n,k)*binomial(2k,k))^2, where e = (d-c)/4: A002894 (c=0,d=4,e=1), A246467 (c=1,d=5,e=1), A246876 (c=2,d=6,e=1), A246906 (c=3,d=7,e=1), A307811 (c=5,d=9,e=1), A322240 (c=-3,d=5,e=2), A322243 (c=-1,d=7,e=2), A246923 (c=1,d=9,e=2), A248167 (c=3, d=11,e=2), A322247 (c=-1,d=11,e=3), this sequence (c=4,d=16,e=3), A322245 (c=-5,d=11,e=4), A322249 (c=-3,d=13,e=4).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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