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%I #11 Mar 06 2022 08:30:54
%S 0,1,7,42,230,1190,5922,28644,135564,630630,2892890,13117676,58903572,
%T 262303132,1159666900,5094808200,22259364120,96773942790,418882316490,
%U 1805951924700,7758285404100,33221013445620,141830949914940,603876402587640,2564713671647400
%N a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2.
%F G.f.: x * (1 - x) * (1 - 2*x) / (1 - 4*x)^(5/2).
%F E.g.f.: x * exp(2*x) * (2 * (1 + x) * BesselI(0,2*x) + (1 + 2*x) * BesselI(1,2*x)) / 2.
%F a(n) = n * ((n+1)^2 - 2) * Catalan(n-1) / 2.
%F a(n) = Sum_{k=0..n} binomial(n,k)^2 * A000217(k).
%F a(n) ~ 2^(2*n-3) * n^(3/2) / sqrt(Pi).
%F D-finite with recurrence (n-1)*(n^2-2)*a(n) -2*(2*n-3)*(n^2+2*n-1)*a(n-1)=0. - _R. J. Mathar_, Mar 06 2022
%t Table[((n + 1)^2 - 2) Binomial[2 n - 2, n - 1]/2, {n, 0, 24}]
%t nmax = 24; CoefficientList[Series[x (1 - x) (1 - 2 x)/(1 - 4 x)^(5/2), {x, 0, nmax}], x]
%o (PARI) a(n) = ((n+1)^2 - 2) * binomial(2*n-2,n-1) / 2 \\ _Andrew Howroyd_, Nov 20 2021
%Y Cf. A000108, A000217, A000984, A001700, A001793, A002457, A002544, A008865, A037966, A088218, A127736.
%K nonn,easy
%O 0,3
%A _Ilya Gutkovskiy_, Nov 17 2021
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