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A360317 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

%I A360317 #22 Aug 22 2025 11:02:45
%S A360317 1,2,10,52,278,1516,8388,46920,264678,1503052,8581676,49215256,
%T A360317 283297660,1635904376,9472214344,54975423504,319729353606,
%U A360317 1862896455180,10871759717916,63539265366264,371837338366740,2178604586281128,12778264475444280,75022726995053808
%N A360317 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).
%F A360317 G.f.: sqrt( (1-2*x)/(1-6*x) ).
%F A360317 n*a(n) = 2*(4*n-3)*a(n-1) - 12*(n-2)*a(n-2).
%F A360317 Sum_{i=0..n} Sum_{j=0..i} (1/2)^i * a(j) * a(i-j) = 3^n.
%F A360317 a(n) = 2 * A005573(n-1) for n > 0.
%F A360317 a(n) ~ 2^(n + 1/2) * 3^(n - 1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Feb 04 2023
%F A360317 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A360317 a(n) = (1/2)^n * Sum_{k=0..n} 3^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A360317 a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)
%o A360317 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
%o A360317 (PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-2*x)/(1-6*x)))
%Y A360317 Cf. A063886, A085362, A360318, A360319, A360321, A360322.
%Y A360317 Cf. A005573, A085363.
%K A360317 nonn,easy,changed
%O A360317 0,2
%A A360317 _Seiichi Manyama_, Feb 03 2023