A246467
G.f.: 1 / AGM(1-5*x, sqrt((1-x)*(1-25*x))).
Original entry on oeis.org
1, 9, 121, 2025, 38025, 762129, 15912121, 341621289, 7484845225, 166549691025, 3751508008161, 85341068948529, 1957289174870121, 45199191579030225, 1049893021288265625, 24510327614556266025, 574726636455361317225, 13528549573868347823025, 319541915502909478890625
Offset: 0
G.f.: A(x) = 1 + 9*x + 121*x^2 + 2025*x^3 + 38025*x^4 + 762129*x^5 +...
where the square-root of the terms yields A026375:
[1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, ...]
the g.f. of which is 1/sqrt((1-x)*(1-5*x)).
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CoefficientList[Series[1/ArithmeticGeometricMean[1-5x,Sqrt[(1-x)(1-25x)]],{x,0,20}],x] (* Harvey P. Dale, Nov 01 2023 *)
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{a(n)=polcoeff( 1 / agm(1-5*x, sqrt((1-x)*(1-25*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=sum(k=0,n,binomial(n,k)*binomial(2*k,k))^2}
for(n=0, 20, print1(a(n), ", "))
A307810
Expansion of 1/AGM(1-64*x, sqrt((1-16*x)*(1-256*x))).
Original entry on oeis.org
1, 100, 13924, 2371600, 453093796, 92598490000, 19745403216400, 4333667896360000, 971177275449892900, 221106619001508490000, 50967394891692703241104, 11866732390447357481358400, 2785834789480617203561744656, 658549235163074008904405646400
Offset: 0
(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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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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N=20; x='x+O('x^N); Vec(1/agm(1-64*x, sqrt((1-16*x)*(1-256*x))))
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{a(n) = sum(k=0, n, 4^(n-k)*3^k*binomial(n, k)*binomial(2*k, k))^2}
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{a(n) = sum(k=0, n, 16^(n-k)*(-3)^k*binomial(n, k)*binomial(2*k, k))^2}
A307811
Expansion of 1/AGM(1-45*x, sqrt((1-25*x)*(1-81*x))).
Original entry on oeis.org
1, 49, 2601, 148225, 8970025, 570111129, 37678303881, 2567836387809, 179267329355625, 12754414737348025, 921185098227422161, 67340346156989933769, 4971327735657992896201, 369994703739586257235225, 27725052308247030792515625, 2089567204521186409129541025
Offset: 0
(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), this sequence (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),
A307810 (c=4,d=16,e=3),
A322245 (c=-5,d=11,e=4),
A322249 (c=-3,d=13,e=4).
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a[n_] := Sum[5^(n-k) * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}]^2; Array[a, 16, 0] // Flatten (* Amiram Eldar, May 13 2021 *)
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N=20; x='x+O('x^N); Vec(1/agm(1-45*x, sqrt((1-25*x)*(1-81*x))))
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{a(n) = sum(k=0, n, 5^(n-k)*binomial(n, k)*binomial(2*k, k))^2}
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{a(n) = sum(k=0, n, 9^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))^2}
Showing 1-3 of 3 results.
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