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A322240
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a(n) = A084605(n)^2, the square of the central coefficient in (1 + x + 4*x^2)^n.
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5
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1, 1, 81, 625, 21025, 314721, 8071281, 155975121, 3685097025, 79065004225, 1832690505361, 41201108978481, 955863424195681, 22002282841672225, 513656632397480625, 11987246142597870225, 281793325930991748225, 6636532731958908324225, 156986802654088221500625, 3721955785026092495600625, 88512056987508595608293025
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OFFSET
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0,3
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COMMENTS
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The g.f. of A084605 is 1/sqrt(1 - 2*x - 15*x^2).
Ignoring initial term, a(n) equals the logarithmic derivative of A322241.
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LINKS
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FORMULA
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G.f.: 1 / AGM(1 + 15*x, sqrt((1 - 9*x)*(1 - 25*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
G.f.: 1 / AGM((1-3*x)*(1-5*x), (1+3*x)*(1+5*x)) = Sum_{n>=0} a(n) * x^(2*n).
a(n) = A084605(n)^2, where A084605(n) = Sum_{k=0..n} (-3)^(n-k) * 2^k * binomial(n,k)*binomial(2*k,k).
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EXAMPLE
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G.f.: A(x) = 1 + x + 81*x^2 + 625*x^3 + 21025*x^4 + 314721*x^5 + 8071281*x^6 + 155975121*x^7 + 3685097025*x^8 + 79065004225*x^9 + ... + A084605(n)^2*x^n + ...
that is,
A(x) = 1 + x + 9^2*x^2 + 25^2*x^3 + 145^2*x^4 + 561^2*x^5 + 2841^2*x^6 + 12489^2*x^7 + 60705^2*x^8 + 281185^2*x^9 + ... + A084605(n)^2*x^n + ...
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MATHEMATICA
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a[n_] := Sum[(-3)^(n-k) * 2^k * Binomial[n, k] * Binomial[2k, k], {k, 0, n}]^2; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)
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PROG
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{a(n)=polcoeff(1/sqrt(1 - 2*x - 15*x^2 +x*O(x^n)), n)^2}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Using AGM: */
{a(n)=polcoeff( 1 / 1 / agm(1 + 3*5*x, sqrt((1 - 3^2*x)*(1 - 5^2*x) +x*O(x^n))), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Square of binomial sum */
{a(n) = sum(k=0, n, (-3)^(n-k)*2^k*binomial(n, k)*binomial(2*k, k))^2}
for(n=0, 20, print1(a(n), ", "))
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CROSSREFS
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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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