A162746 - OEIS

%I #10 Oct 05 2023 10:59:44

%S 1,2,5,14,42,132,427,1402,4629,15290,50412,165816,544253,1783602,

%T 5839313,19106766,62504002,204457540,668825279,2188016442,7158417217,

%U 23421034442,76632061852,250740203864,820430583305,2684486330562,8783760256301,28740810537518,94040879244602

%N Row sums of Fibonacci-Pascal triangle A162745.

%C Second binomial transform of aerated Fibonacci numbers.

%C Hankel transform is 1,1,1,-1,0,0,0,...

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-23,28,-11).

%F G.f.: (1-2x)^3/(1-8x+23x^2-28x^3+11x^4);

%F a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*2^(n-2k)*F(k+1).

%F a(n) = Sum_{k=0..n} C(n,k)*2^(n-k)*F(k/2+1)*(1+(-1)^k)/2.

%t LinearRecurrence[{8,-23,28,-11},{1,2,5,14},30] (* _Harvey P. Dale_, Oct 05 2023 *)

%o (PARI) T(n,k) = sum(j=0, n, binomial(n,j)*binomial(n-j,2*(k-j))*fibonacci(k-j+1));

%o a(n) = vecsum(vector(n+1, k, T(n, k-1))); \\ _Michel Marcus_, Nov 11 2022

%Y Cf. A162745.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 12 2009