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A071722 Expansion of (1+x^2*C^2)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

%I A071722 #12 Dec 02 2024 08:02:20
%S A071722 1,3,10,33,110,372,1276,4433,15574,55250,197676,712538,2585292,
%T A071722 9434830,34610400,127553745,472055910,1753616370,6536826780,
%U A071722 24443315550,91664179620,344655239760,1299052403688,4907335827258,18576824685820
%N A071722 Expansion of (1+x^2*C^2)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%F A071722 Conjecture: 2*(n+3)*a(n) + 4*(-3*n-4)*a(n-1) + (17*n-9)*a(n-2) + 2*(-2*n+5)*a(n-3) = 0. - _R. J. Mathar_, Aug 25 2013
%F A071722 a(n) = (3*binomial(2*n+2, n) + 5*binomial(2*n, n+2))/(n + 3). - _Tani Akinari_, Dec 01 2024
%p A071722 a := n -> (17*n^2 + 13*n + 6)*binomial(2*n, n)/((n + 1)*(n + 2)*(n + 3)): seq(a(n), n = 0..24);  # _Peter Luschny_, Dec 01 2024
%o A071722 (Maxima) a(n):=(3*binomial(2*n+2,n)+5*binomial(2*n,n+2))/(n+3); makelist(a(n),n,0,50);
%o A071722 /* _Tani Akinari_, Dec 01 2024 */
%Y A071722 gf=(1+x^2*C^2)*C^m: A000782 (m=1), A071721 (m=2), this sequence (m=3), A071723 (m=4).
%Y A071722 Cf. A000108.
%K A071722 nonn
%O A071722 0,2
%A A071722 _N. J. A. Sloane_, Jun 06 2002