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A360318 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

%I A360318 #19 Aug 22 2025 11:02:27
%S A360318 1,2,12,74,466,2982,19320,126390,833220,5527190,36852052,246751854,
%T A360318 1658106394,11176100138,75528743352,511600414554,3472363279170,
%U A360318 23609924743590,160788499672020,1096566516149790,7488135911236806,51193972101241362,350368409215623192
%N A360318 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).
%F A360318 G.f.: sqrt( (1-3*x)/(1-7*x) ).
%F A360318 n*a(n) = 2*(5*n-4)*a(n-1) - 21*(n-2)*a(n-2).
%F A360318 Sum_{i=0..n} Sum_{j=0..i} (1/3)^i * a(j) * a(i-j) = (7/3)^n.
%F A360318 a(n) = 2 * A122898(n-1) for n > 0.
%F A360318 a(n) ~ 2 * 7^(n - 1/2) / sqrt(Pi*n). - _Vaclav Kotesovec_, Feb 04 2023
%F A360318 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A360318 a(n) = (1/4)^n * Sum_{k=0..n} 7^k * 3^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A360318 a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)
%o A360318 (PARI) a(n) = sum(k=0, n, 3^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
%o A360318 (PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1-3*x)/(1-7*x)))
%Y A360318 Cf. A063886, A085362, A360317, A360319, A360321, A360322.
%Y A360318 Cf. A085364, A122898.
%K A360318 nonn,easy,changed
%O A360318 0,2
%A A360318 _Seiichi Manyama_, Feb 03 2023