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From Peter Bala, Apr 29 2024: (Start)
a(n) = Sum_{k = 0..n} binomial(n, 2*k)*Catalan(k)*4^(n-k-1). - _Peter Bala_, Apr 29 2024
O.g.f.: A(x) = (1 - 4*x - 8*x^2 - sqrt(1 - 8*x))/(32*x^2).
A(x) = series reversion of x*c(-x)/(1 + 4*x), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and c(-x)/(1 + 4*x) is the g.f. of (-1)^n*A000346(n). (End)
a(n) = Sum_{k = 0..n} binomial(n, 2*k)*Catalan(k)*4^(n-k-1). - Peter Bala, Apr 29 2024
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D-finite with recurrence: 4*(2*n+1)*a(n-1) - (n+2)*a(n) = 0, a(1) = 1. - Georg Fischer, May 23 2021
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