A361758 a(n) = [x^n] (x^5 + 5*x^4 + 4*x^3 - 3*x + 1)/((1 - x)*(x^2 + 2*x - 1)^2).

1, 2, 4, 10, 30, 90, 264, 754, 2106, 5778, 15628, 41786, 110678, 290858, 759312, 1971042, 5091442, 13095586, 33555988, 85695978, 218198158, 554081146, 1403588376, 3547702610, 8949110954, 22532603954, 56637795100, 142141826074, 356212187334, 891481312842

Offset: 0

Author

Peter Luschny, Mar 23 2023

Keywords

• nonn

Crossrefs

Cf. A361745.

Programs

Maple

a := proc(n) option remember; if n < 4 then return [1, 2, 4, 10][n + 1] fi;
((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);
# Alternative:
ogf := (x^5 + 5*x^4 + 4*x^3 - 3*x + 1)/((1 - x)*(x^2 + 2*x - 1)^2):
ser := series(ogf, x, 40): seq(coeff(ser, x, n), n = 0..29);

Formula

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.