A270737 - OEIS

%I #16 May 10 2020 15:45:04

%S 1,3,5,10,20,42,91,195,415,880,1864,3952,8385,17795,37765,80138,

%T 170044,360810,765595,1624515,3447071,7314368,15520400,32932800,

%U 69880225,148279107,314634021,667623210,1416632420,3005958090,6378354619

%N a(n) = ((n+2)/2)*Sum_{k=0..n/2} (Sum_{i=0..n-2*k} (binomial(k+1,n-2*k-i)*binomial(k+i,k))*F(k+1)/(k+1)), where F(k) is Fibonacci numbers.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2, 0, 0, 0, 2, 1).

%F G.f.: (-x^2+x+1)/(-x^6-2*x^5-2*x+1).

%F a(n) = 2*a(n-1) + 2*a(n-5) + a(n-6). - _G. C. Greubel_, mar 25 2016

%t Table[((n + 2)/2) Sum[Sum[(Binomial[k + 1, n - 2 k - i] Binomial[k + i, k])/(k + 1) Fibonacci[k + 1], {i, 0, n - 2 k}], {k, 0, n/2}], {n, 0, 30}] (* or *)

%t CoefficientList[Series[(-x^2 + x + 1)/(-x^6 - 2 x^5 - 2 x + 1), {x, 0, 30}], x] (* _Michael De Vlieger_, Mar 25 2016 *)

%t LinearRecurrence[{2, 0, 0, 0, 2, 1}, {1, 3, 5, 10, 20, 42}, 100] (* _G. C. Greubel_, Mar 25 2016 *)

%o (Maxima)

%o a(n):=(n+2)/2*(sum(sum(binomial(k+1,n-2*k-i)*binomial(k+i,k),i,0,n-2*k)*fib(k+1)/(k+1),k,0,n/2));

%o (PARI) x='x+O('x^200); Vec((-x^2+x+1)/(-x^6-2*x^5-2*x+1)) \\ _Altug Alkan_, Mar 22 2016

%Y Cf. A000045, A113413.

%K nonn

%O 0,2

%A _Vladimir Kruchinin_, Mar 22 2016