A332698 a(n) = (8*n^3 + 15*n^2 + 13*n)/6.

0, 6, 25, 65, 134, 240, 391, 595, 860, 1194, 1605, 2101, 2690, 3380, 4179, 5095, 6136, 7310, 8625, 10089, 11710, 13496, 15455, 17595, 19924, 22450, 25181, 28125, 31290, 34684, 38315, 42191, 46320, 50710, 55369, 60305, 65526, 71040, 76855, 82979, 89420, 96186

Offset: 0

Author

Peter Luschny, Feb 20 2020

Keywords

• nonn
• easy

Comments

The end values of the partial rows on the main diagonal of A332662 in the representation in the example section.

Links

• Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Crossrefs

Cf. A332662, A331987.

Programs

Maple

a := n -> (8*n^3 + 15*n^2 + 13*n)/6: seq(a(n), n=0..41);
gf := (x*(x^2 + x + 6))/(x - 1)^4: ser := series(gf, x, 44):
seq(coeff(ser, x, n), n=0..41);

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {0, 6, 25, 65}, 42]
Table[(8n^3+15n^2+13n)/6,{n,0,50}] (* Harvey P. Dale, Sep 13 2024 *)

Formula

a(n) = [x^n] (x*(x^2 + x + 6))/(x - 1)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

a(n) = binomial(n+2, 3) + binomial(n+1, 3) + 2*(n+1)*binomial(n+1, 2) + binomial(n, 1) = A331987(n) + n.