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%I #10 Dec 13 2018 08:04:20
%S 1,25,1849,156025,14523721,1426950625,145317252025,15178231605625,
%T 1615509001626025,174471431239950625,19062335608125901729,
%U 2102483602307417980225,233721380163477368733481,26154175972512598202392225,2943361280244176889333396889,332869229155486455718147125625,37806108834415039621850996946025,4310099976506176089944803738530625,493021434686696395739629566004713025
%N a(n) = A322246(n)^2, the square of the central coefficient in (1 + 5*x + 9*x^2)^n.
%H Seiichi Manyama, <a href="/A322247/b322247.txt">Table of n, a(n) for n = 0..481</a>
%F G.f.: 1 / AGM(1 + 11*x, sqrt((1 - x)*(1 - 11^2*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
%F G.f.: 1 / AGM((1-x)*(1-11*x), (1+x)*(1+11*x)) = Sum_{n>=0} a(n)*x^(2*n).
%F a(n) = A322248(n)^2, where A322248(n) = a(n) = Sum_{k=0..n} (-1)^(n-k) * 3^k * binomial(n,k)*binomial(2*k,k).
%F a(n) ~ 11^(2*n + 1) / (12*Pi*n). - _Vaclav Kotesovec_, Dec 13 2018
%e G.f.: A(x) = 1 + 25*x + 1849*x^2 + 156025*x^3 + 14523721*x^4 + 1426950625*x^5 + 145317252025*x^6 + 15178231605625*x^7 + 1615509001626025*x^8 + ...
%e that is,
%e A(x) = 1 + 5^2*x + 43^2*x^2 + 395^2*x^3 + 3811^2*x^4 + 37775^2*x^5 + 381205^2*x^6 + 3895925^2*x^7 + 40193395^2*x^8 + 417697775^2*x^9 + ... + A322246(n)^2*x^n + ...
%t a[n_] := Sum[(-1)^(n-k) * 3^k * Binomial[n,k] * Binomial[2k,k], {k, 0, n}]^2; Array[a, 20, 0] (* _Amiram Eldar_, Dec 13 2018 *)
%o (PARI) /* a(n) = A322246(n)^2 - g.f. */
%o {a(n)=polcoeff(1/sqrt( (1 - x)*(1 + 11*x) +x*O(x^n)), n)^2}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) /* a(n) = A322246(n)^2 - g.f. */
%o {a(n) = polcoeff( (1 + 5*x + 9*x^2 +x*O(x^n))^n, n)^2}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) /* a(n) = A322246(n)^2 - binomial sum */
%o {a(n) = sum(k=0,n, (-1)^(n-k)*3^k*binomial(n,k)*binomial(2*k,k))^2}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) /* a(n) = A322246(n)^2 - binomial sum */
%o {a(n) = sum(k=0,n, 11^(n-k)*(-3)^k*binomial(n,k)*binomial(2*k,k))^2}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) /* Using AGM: */
%o {a(n)=polcoeff( 1 / agm(1 + 1*11*x, sqrt((1 - 1^2*x)*(1 - 11^2*x) +x*O(x^n))), n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A246467, A246906, A246923, A322246.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 10 2018
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