OFFSET
0,3
FORMULA
G.f.: (1 - sqrt(1 - 4*x))/(sqrt(1 - 4*x)*(x^2 - x) + x^2 - 3*x + 2).
a(n) ~ 2^(2*n + 4) / (49 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Sep 26 2018
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
MAPLE
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
MATHEMATICA
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 *)
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 *)
PROG
(Maxima) a(n):=sum(binomial(2*n-3*k+1, n-k)*k/(n-k+1), k, 1, n);
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Sep 25 2018
STATUS
approved