A270737 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.

1, 3, 5, 10, 20, 42, 91, 195, 415, 880, 1864, 3952, 8385, 17795, 37765, 80138, 170044, 360810, 765595, 1624515, 3447071, 7314368, 15520400, 32932800, 69880225, 148279107, 314634021, 667623210, 1416632420, 3005958090, 6378354619

Offset: 0

Author

Vladimir Kruchinin, Mar 22 2016

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,2,1).

Crossrefs

Cf. A000045, A113413.

Programs

Mathematica

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 *)
CoefficientList[Series[(-x^2 + x + 1)/(-x^6 - 2 x^5 - 2 x + 1), {x, 0, 30}], x] (* Michael De Vlieger, Mar 25 2016 *)
LinearRecurrence[{2, 0, 0, 0, 2, 1}, {1, 3, 5, 10, 20, 42}, 100] (* G. C. Greubel, Mar 25 2016 *)

Maxima

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));

PARI

my(x='x+O('x^40)); Vec((-x^2+x+1)/(-x^6-2*x^5-2*x+1)) \\ Altug Alkan, Mar 22 2016

Formula

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

a(n) = 2*a(n-1) + 2*a(n-5) + a(n-6). - G. C. Greubel, Mar 25 2016