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A374487 Expansion of 1/(1 - 2*x - 7*x^2)^(3/2).

%I A374487 #30 Aug 25 2025 05:14:27
%S A374487 1,3,18,70,315,1281,5348,21708,88245,355135,1425270,5692050,22666735,
%T A374487 89986365,356400840,1408459928,5555679849,21877337979,86020384730,
%U A374487 337769595870,1324677499299,5189411915897,20308936981932,79406140870500,310206869770525,1210898719869111
%N A374487 Expansion of 1/(1 - 2*x - 7*x^2)^(3/2).
%H A374487 Paolo Xausa, <a href="/A374487/b374487.txt">Table of n, a(n) for n = 0..1000</a>
%F A374487 a(0) = 1, a(1) = 3; a(n) = ((2*n+1)*a(n-1) + 7*(n+1)*a(n-2))/n.
%F A374487 a(n) = binomial(n+2,2) * A025235(n).
%F A374487 From _Seiichi Manyama_, Aug 20 2025: (Start)
%F A374487 a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 2^k * binomial(n+1,n-2*k) * binomial(2*k+1,k).
%F A374487 a(n) = Sum_{k=0..n} (1/2)^k * (7/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)
%F A374487 a(n) ~ sqrt(n) * (1 + 2*sqrt(2))^(n + 3/2) / (2^(11/4) * sqrt(Pi)). - _Vaclav Kotesovec_, Aug 21 2025
%t A374487 Module[{x}, CoefficientList[Series[1/(1 - (7*x + 2)*x)^(3/2), {x, 0, 25}], x]] (* _Paolo Xausa_, Aug 25 2025 *)
%o A374487 (PARI) a(n) = binomial(n+2, 2)*sum(k=0, n\2, 2^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1));
%Y A374487 Cf. A102839, A102840, A374488.
%Y A374487 Cf. A014431, A025235, A084601.
%K A374487 nonn,changed
%O A374487 0,2
%A A374487 _Seiichi Manyama_, Jul 09 2024