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A270561 Binomial transform(2) of Motzkin numbers.

%I A270561 #27 Oct 30 2017 18:00:20
%S A270561 1,3,11,42,164,649,2592,10423,42140,171133,697641,2853587,11707542,
%T A270561 48166629,198677283,821495226,3404577572,14140959469,58859315929,
%U A270561 245493952745,1025954717376,4295887639272,18021572480109,75740267331717
%N A270561 Binomial transform(2) of Motzkin numbers.
%H A270561 G. C. Greubel, <a href="/A270561/b270561.txt">Table of n, a(n) for n = 0..1000</a>
%F A270561 G.f.: M(A(x))*A(x)/(2*x-A(x)), where M(x) is g.f. of Motzkin numbers (A001006) and A(x)/x is the g.f. of Catalan numbers (A000108).
%F A270561 a(n) = Sum_{i=0..n}((Sum_{k=0..i}((binomial(i,2*k)*binomial(2*k,k))/(k+1)))* binomial(2*n-i,n-i)).
%F A270561 a(n) = Sum_{k=0,n} (T(n,k)*m(k)), where m(k) is Motzkin numbers (A001006), T(n,k) = binomial(2*n-k,n) (triangle A092392).
%F A270561 a(n) ~ 3^(2*n + 5/2) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). - _Vaclav Kotesovec_, Mar 19 2016
%F A270561 a(n) = [x^n] (1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2*(1 - x)^(n+1)). - _Ilya Gutkovskiy_, Oct 30 2017
%t A270561 Table[Sum[Sum[Binomial[i, 2 k] Binomial[2 k, k]/(k + 1), {k, 0, i}] Binomial[2 n - i, n - i], {i, 0, n}], {n, 0, 23}] (* or *)
%t A270561 nn = 23; m = CoefficientList[Series[(1 - x - (1 - 2 x - 3 x^2)^(1/2))/(2 x^2), {x, 0, nn}], x]; Table[Sum[Binomial[2 n - k, n] m[[k + 1]], {k, 0, n}], {n, 0, nn}] (* _Michael De Vlieger_, Mar 19 2016, latter after _Jean-François Alcover_ at A001006 *)
%o A270561 (Maxima)
%o A270561 A(x):=(1-sqrt(1-4*x))/2;
%o A270561 M(x) := ( 1 - x - (1-2*x-3*x^2)^(1/2) ) / (2*x^2);
%o A270561 makelist(coeff(taylor(M(A(x))*A(x)/(2*x-A(x)),x,0,10),x,n),n,0,10);
%o A270561 (Maxima)
%o A270561 a(n):=sum((sum((binomial(i,2*k)*binomial(2*k,k))/(k+1),k,0,i))*binomial(2*n-i,n-i),i,0,n);
%o A270561 (PARI) a(n) = sum(i=0, n, sum(k=0, i, binomial(i, 2*k) * binomial(2*k, k) / (k+1)) * binomial(2*n-i, n-i)); \\ _Indranil Ghosh_, Mar 04 2017
%Y A270561 Cf. A000108, A001006, A092392.
%K A270561 nonn
%O A270561 0,2
%A A270561 _Vladimir Kruchinin_, Mar 19 2016