%I #39 Jan 30 2020 21:29:18

%S 0,1,3,6,11,23,60,182,589,1960,6641,22849,79676,281048,1001100,

%T 3595865,13009663,47366234,173415160,638044198,2357941155,8748646416,

%U 32576869239,121701491725,456012458960,1713339737046,6453584646774,24364925259967,92185136438926,349479503542513

%N a(n) = Sum_{k=1..n} binomial(2*n - 3*k + 1, n - k)*k/(n - k + 1).

%F G.f.: (1 - sqrt(1 - 4*x))/(sqrt(1 - 4*x)*(x^2 - x) + x^2 - 3*x + 2).

%F a(n) ~ 2^(2*n + 4) / (49 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 26 2018

%F D-finite with recurrence: n*a(n) - (5*n - 6)*a(n-1) + 2*(2*n - 3)*a(n-2) + n*a(n-3) - 2*(2*n - 3)*a(n-4) + 3*(n - 2) = 0 for n > 3. - _Bruno Berselli_, Sep 26 2018

%p a:=n->add(binomial(2*n-3*k+1,n-k)*k/(n-k+1),k=1..n): seq(a(n),n=0..30); # _Muniru A Asiru_, Sep 25 2018

%t a[n_] := Sum[Binomial[2 n-3 k + 1, n - k] k/(n - k + 1), {k, 1, n}]; Array[a, 50] (* or *) CoefficientList[Series[(1 - Sqrt[1 - 4 x])/(Sqrt[1 - 4 x] (x^2 - x) + x^2 - 3 x + 2), {x, 0, 50}], x] (* _Stefano Spezia_, Sep 25 2018 *)

%t RecurrenceTable[{n a[n] - (5 n - 6) a[n - 1] + 2 (2 n - 3) a[n - 2] + n a[n - 3] - 2 (2 n - 3) a[n - 4] + 3 (n - 2) == 0, a[0] == 0, a[1] == 1, a[2] == 3, a[3] == 6}, a, {n, 0, 30}] (* _Bruno Berselli_, Sep 26 2018 *)

%o (Maxima) a(n):=sum(binomial(2*n-3*k+1,n-k)*k/(n-k+1),k,1,n);

%o (PARI) x='x+O('x^40); concat(0, Vec((1-sqrt(1-4*x))/(sqrt(1-4*x)*(x^2-x)+x^2-3*x+2))) \\ _Altug Alkan_, Sep 25 2018

%o (GAP) List([0..30], n-> Sum([1..n], k-> Binomial(2*n-3*k+1,n-k)*k/(n-k+1))); # _Muniru A Asiru_, Sep 25 2018

%Y Cf. A000108, A030238.

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Sep 25 2018