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

0,2

Links

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

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

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 *)

Prog

(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

Crossrefs

Cf. A000108, A001006, A092392.

Sequence in context: A032443 A180907 A143464 * A259858 A359711 A117641

Adjacent sequences: A270558 A270559 A270560 * A270562 A270563 A270564

Keyword

nonn

Author

Vladimir Kruchinin, Mar 19 2016

Status

approved