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A322245 a(n) = A322244(n)^2, the square of the central coefficient in (1 + 3*x + 16x^2)^n.

%I A322245 #13 Dec 13 2018 08:43:20
%S A322245 1,9,1681,99225,11189025,1019077929,108167974321,11137724806329,
%T A322245 1197175570050625,129043286745662025,14139849178569444561,
%U A322245 1560056939689534791129,173632800433761049827681,19441545948416137605093225,2189471746316530595782640625,247739375533243457859858956025,28151322919072607132343448452225,3210745922813847247161044673062025
%N A322245 a(n) = A322244(n)^2, the square of the central coefficient in (1 + 3*x + 16x^2)^n.
%C A322245 The g.f. of A322244 is 1/sqrt(1 - 6*x - 55*x^2).
%H A322245 Seiichi Manyama, <a href="/A322245/b322245.txt">Table of n, a(n) for n = 0..481</a>
%F A322245 G.f.: 1 / AGM(1 + 5*11*x, sqrt((1 - 5^2*x)*(1 - 11^2*x)) ), where AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) is the arithmetic-geometric mean.
%F A322245 G.f.: 1 / AGM((1-5*x)*(1-11*x), (1+5*x)*(1+11*x)) = Sum_{n>=0} a(n)*x^(2*n).
%F A322245 a(n) = A322244(n)^2, where A322244(n) = a(n) = Sum_{k=0..n} 11^(n-k) * (-4)^k * binomial(n,k)*binomial(2*k,k).
%F A322245 a(n) ~ 11^(2*n + 1) / (16*Pi*n). - _Vaclav Kotesovec_, Dec 13 2018
%e A322245 G.f.: A(x) = 1 + 9*x + 1681*x^2 + 99225*x^3 + 11189025*x^4 + 1019077929*x^5 + 108167974321*x^6 + 11137724806329*x^7 + 1197175570050625*x^8 + ...
%e A322245 that is,
%e A322245 A(x) = 1 + 3^2*x + 41^2*x^2 + 315^2*x^3 + 3345^2*x^4 + 31923^2*x^5 + 328889^2*x^6 + 3337323^2*x^7 + 34600225^2*x^8 + ... + A322244(n)^2*x^n + ...
%t A322245 a[n_] := Sum[11^(n-k) * (-4)^k * Binomial[n,k] * Binomial[2k,k], {k, 0, n}]^2; Array[a, 20, 0] (* _Amiram Eldar_, Dec 13 2018 *)
%o A322245 (PARI) /* a(n) = A322244(n)^2 - g.f. */
%o A322245 {a(n)=polcoeff(1/sqrt(1 - 6*x - 55*x^2 +x*O(x^n)), n)^2}
%o A322245 for(n=0, 20, print1(a(n), ", "))
%o A322245 (PARI) /* a(n) = A322244(n)^2 - binomial sum */
%o A322245 {a(n) = sum(k=0,n, 11^(n-k)*(-4)^k*binomial(n,k)*binomial(2*k,k))^2}
%o A322245 for(n=0, 20, print1(a(n), ", "))
%o A322245 (PARI) /* Using AGM: */
%o A322245 {a(n)=polcoeff( 1 / agm(1 + 5*11*x, sqrt((1 - 5^2*x)*(1 - 11^2*x) +x*O(x^n))), n)}
%o A322245 for(n=0, 20, print1(a(n), ", "))
%Y A322245 Cf. A322244.
%K A322245 nonn
%O A322245 0,2
%A A322245 _Paul D. Hanna_, Dec 09 2018