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A112328 a(n) = (n+1)*binomial(2n+2,n+1)-3*4^n+binomial(2n,n).

%I A112328 #20 Jan 21 2025 03:26:00
%S A112328 2,18,108,562,2724,12660,57240,253842,1109748,4798780,20572392,
%T A112328 87580308,370706408,1561573032,6551178288,27387484242,114146434068,
%U A112328 474476717292,1967642119368,8142727008732,33634295542968,138696447565272,571069686589008,2348087834494932
%N A112328 a(n) = (n+1)*binomial(2n+2,n+1)-3*4^n+binomial(2n,n).
%C A112328 Row sums of A112327.
%H A112328 Robert Israel, <a href="/A112328/b112328.txt">Table of n, a(n) for n = 1..1656</a>
%H A112328 Frank Ruskey, <a href="https://doi.org/10.1137/0601007">Average shape of binary trees</a>, SIAM J. Alg. Disc. Meth., 1, 1980, 43-50 (Eq. (8)).
%F A112328 G.f.: 4*z*(2-sqrt(1-4*z))/((1-4*z)^(3/2)*(1+sqrt(1-4*z))).
%F A112328 32*(2*n^2 - 9*n + 10)*a(n - 3) - 8*(2*n^2 - 14*n + 15)*a(n - 2) - 2*(2*n^2 + 3*n - 5)*a(n - 1) + n*(n - 1)*a(n) = 0. - _Robert Israel_, Sep 19 2019
%p A112328 a:=n->(n+1)*binomial(2*n+2,n+1)-3*4^n+binomial(2*n,n): seq(a(n),n=1..25);
%t A112328 a[n_]:=(n+1)*Binomial[2n+2,n+1]-3*4^n+Binomial[2n,n]; Array[a,22] (* _Stefano Spezia_, Sep 20 2019 *)
%o A112328 (PARI) a(n) = (n+1)*binomial(2*n+2,n+1)-3*4^n+binomial(2*n,n); \\ _Michel Marcus_, Sep 20 2019
%o A112328 (Magma) [(n+1)*Binomial(2*n+2,n+1)-3*4^n+Binomial(2*n,n):n in [1..22]]; // _Marius A. Burtea_, Sep 20 2019
%Y A112328 Cf. A112327.
%K A112328 nonn,easy
%O A112328 1,1
%A A112328 _Emeric Deutsch_, Sep 04 2005