A361758 - OEIS

%I #4 Mar 23 2023 16:53:27

%S 1,2,4,10,30,90,264,754,2106,5778,15628,41786,110678,290858,759312,

%T 1971042,5091442,13095586,33555988,85695978,218198158,554081146,

%U 1403588376,3547702610,8949110954,22532603954,56637795100,142141826074,356212187334,891481312842

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

%F a(n) = ((3*n^2 - 7*n + 2)*a(n - 1) - (n^2 - n)*a(n - 3) - (n^2 - 3*n)*a(n - 2)) / ((n - 1)*(n - 2)) for n >= 4.

%p a := proc(n) option remember; if n < 4 then return [1, 2, 4, 10][n + 1] fi;

%p ((3*n^2 - 7*n + 2)*a(n - 1) - (n^2 - n)*a(n - 3) - (n^2 - 3*n)*a(n - 2))/((n - 1)*(n - 2)) end: seq(a(n), n = 0..29);

%p # Alternative:

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

%p ser := series(ogf, x, 40): seq(coeff(ser, x, n), n = 0..29);

%Y Cf. A361745.

%K nonn

%O 0,2

%A _Peter Luschny_, Mar 23 2023