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%I #17 Dec 02 2024 08:03:23
%S 1,4,15,54,192,682,2431,8710,31382,113696,414086,1515516,5571750,
%T 20569590,76228095,283481670,1057628550,3957577800,14849601090,
%U 55859886420,210622646520,795898303668,3013646759910,11432740177564,43448822603452,165396657221152
%N Expansion of (1+x^2*C^2)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%F a(n) = (Sum_{k=0..n} (k+1)*(k^2+k+1)*binomial(2*n-k,n))/(n+1). - _Vladimir Kruchinin_, Sep 28 2011
%F a(n) = (4*binomial(2*n+3,n)+6*binomial(2*n+1,n+3))/(n+4). - _Tani Akinari_, Dec 01 2024
%p a := n -> (2*(2*n + 1)*(11*n^2 + 17*n + 12)*binomial(2*n, n))/((n + 1)*(n + 2)*(n + 3)*(n + 4)): seq(a(n), n = 0..25); # _Peter Luschny_, Dec 01 2024
%o (Maxima) a(n):=sum((k+1)*(k^2+k+1)*binomial(2*n-k,n),k,0,n)/(n+1); /* _Vladimir Kruchinin_, Sep 28 2011 */
%o (Maxima) a(n):=(4*binomial(2*n+3,n)+6*binomial(2*n+1,n+3))/(n+4); /* _Tani Akinari_, Dec 01 2024 */
%Y gf=(1+x^2*C^2)*C^m: A000782 (m=1), A071721 (m=2), A071722 (m=3), this sequence (m=4).
%Y Cf. A000108.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jun 06 2002