A270561 Binomial transform(2) of Motzkin numbers.

1, 3, 11, 42, 164, 649, 2592, 10423, 42140, 171133, 697641, 2853587, 11707542, 48166629, 198677283, 821495226, 3404577572, 14140959469, 58859315929, 245493952745, 1025954717376, 4295887639272, 18021572480109, 75740267331717

Offset: 0

Author

Vladimir Kruchinin, Mar 19 2016

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A001006, A092392.

Programs

Mathematica

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 *)
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 *)

Maxima

A(x):=(1-sqrt(1-4*x))/2;
M(x) := ( 1 - x - (1-2*x-3*x^2)^(1/2) ) / (2*x^2);
makelist(coeff(taylor(M(A(x))*A(x)/(2*x-A(x)),x,0,10),x,n),n,0,10);

Maxima

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);

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

Formula

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).

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)).

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).

a(n) ~ 3^(2*n + 5/2) / (sqrt(Pi) * n^(3/2) * 2^(n + 1/2)). - Vaclav Kotesovec, Mar 19 2016

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