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 A270561 Binomial transform(2) of Motzkin numbers. 1
 1, 3, 11, 42, 164, 649, 2592, 10423, 42140, 171133, 697641, 2853587, 11707542, 48166629, 198677283, 821495226, 3404577572, 14140959469, 58859315929, 245493952745, 1025954717376, 4295887639272, 18021572480109, 75740267331717 (list; graph; refs; listen; history; text; internal format)
 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 A117641 A200030 Adjacent sequences:  A270558 A270559 A270560 * A270562 A270563 A270564 KEYWORD nonn AUTHOR Vladimir Kruchinin, Mar 19 2016 STATUS approved

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Last modified October 23 06:21 EDT 2018. Contains 316519 sequences. (Running on oeis4.)