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%I #17 Jul 10 2021 04:42:09
%S 1,3,11,47,219,1075,5459,28383,150131,804515,4355163,23768079,
%T 130572363,721247571,4002344355,22296869823,124633584099,698707769923,
%U 3927060020651,22121780745711,124865811262139,706065855417203,3998950848888051
%N Sum_{k=0..n} ((k+1)*binomial(n+1,k)*binomial(2*n-k,n))/(n+1).
%H D. Drake, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Drake/drake.html">Bijections from Weighted Dyck Paths to Schröder Paths</a, J. Int. Seq. 13 (2010) # 10.9.2, Table 1.
%F G.f.: (-2*x^2+7*x-1)/(2*x*sqrt(x^2-6*x+1))+1/(2*x)-1.
%F G.f. satisfies -A'(x)/A(x)+A'(x)/x, where A(x)/x is g.f. of A155069
%F -(n+1)*(2*n^2+5*n-6)*a(n) +6*(2*n^3+6*n^2-11*n+4)*a(n-1) -(n-2)*(2*n^2+9*n+1)*a(n-2)=0. - _R. J. Mathar_, Jul 21 2017
%F a(n) ~ (1 + sqrt(2))^(2*n) / (2^(5/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 10 2021
%t Table[Sum[(k + 1) Binomial[n + 1, k] Binomial[2 n - k, n]/(n + 1), {k,
%t 0, n}], {n, 0, 22}] (* _Michael De Vlieger_, Sep 26 2015 *)
%o (Maxima)
%o A(x):=x*(3-x-sqrt(1-6*x+x^2))/2;
%o taylor(-diff(A(x),x)/A(x)+diff(A(x),x,1)/x,x,0,27);
%Y Cf. A155069.
%K nonn
%O 0,2
%A _Vladimir Kruchinin_, Sep 26 2015
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