A361732 - OEIS

%I #13 Apr 19 2024 15:34:30

%S 1,1,2,6,20,60,174,490,1352,3672,9850,26158,68892,180180,468454,

%T 1211730,3120400,8004144,20460402,52139990,132502180,335882988,

%U 849507230,2144114234,5401408344,13583493000,34105191146,85504030974,214070361260,535269125508,1336814464470

%N a(n) = [x^n] (x^5 + 5*x^4 + 4*x^3 - 3*x + 1)/(x^2 + 2*x - 1)^2.

%F a(n) = (n*(n - 1)*a(n-2) + 2*n*(n - 2)*a(n-1)) / ((n - 2)*(n - 1)) for n >= 4.

%F a(n) = Sum_{k=0..n-1} F(n-1, 2) for n >= 2, where F(n, x) is the n-th Fibonacci polynomial.

%F a(n) = n*A000129(n-1), a(0)=1, a(1)=1. - _Vladimir Kruchinin_, Apr 19 2024

%F a(n) = 2^(n-2)*n*hypergeom([(3-n)/2, (2-n)/2], [2-n], -1)) for n >= 2. - _Peter Luschny_, Apr 19 2024

%p a := proc(n) option remember; if n < 4 then return [1, 1, 2, 6][n + 1] fi;

%p (n*(n - 1)*a(n - 2) + 2*n*(n - 2)*a(n - 1)) / ((n - 2)*(n - 1)) end:

%p seq(a(n), n = 0..30);

%p # Alternative:

%p F := n -> add(combinat:-fibonacci(n - 1, 2), k = 0..n-1):

%p a := n -> F(n) + ifelse(n < 2, 1, 0): seq(a(n), n=0..30);

%p # Using the generating function:

%p ogf := (x^5 + 5*x^4 + 4*x^3 - 3*x + 1)/(x^2 + 2*x - 1)^2:

%p ser := series(ogf, x, 40): seq(coeff(ser, x, n), n = 0..30);

%p # Or:

%p a := n -> ifelse(n < 2, 1, 2^(n-2)*n*hypergeom([(3-n)/2, (2-n)/2], [2-n], -1));

%p seq(simplify(a(n)), n = 0..30);  # _Peter Luschny_, Apr 19 2024

%Y Cf. A000129, A361758.

%K nonn

%O 0,3

%A _Peter Luschny_, Mar 23 2023