A319636 a(n) = Sum_{k=1..n} binomial(2*n - 3*k + 1, n - k)*k/(n - k + 1).

0, 1, 3, 6, 11, 23, 60, 182, 589, 1960, 6641, 22849, 79676, 281048, 1001100, 3595865, 13009663, 47366234, 173415160, 638044198, 2357941155, 8748646416, 32576869239, 121701491725, 456012458960, 1713339737046, 6453584646774, 24364925259967, 92185136438926, 349479503542513

Offset: 0

Author

Vladimir Kruchinin, Sep 25 2018

Keywords

• nonn

Crossrefs

Cf. A000108, A030238.

Programs

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

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 *)

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

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