A370912 a(n) = n*(n + 2)*(n + 4).

0, 15, 48, 105, 192, 315, 480, 693, 960, 1287, 1680, 2145, 2688, 3315, 4032, 4845, 5760, 6783, 7920, 9177, 10560, 12075, 13728, 15525, 17472, 19575, 21840, 24273, 26880, 29667, 32640, 35805, 39168, 42735, 46512, 50505, 54720, 59163, 63840, 68757, 73920

Offset: 0

Author

Peter Luschny, Mar 05 2024

Keywords

• nonn
• easy

Links

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

Crossrefs

Cases of A370419(n, k): A000012 (n=0), A001477 (n=1), A005563 (n=2), this sequence (n=3), A190577(n=4).

Cf. A077415, A370890.

Cf. A028347, A028387, A211441.

Programs

Maple

a := n -> n*(n + 2)*(n + 4): seq(a(n), n = 0..40);
# Using the generating function:
gf := 3*x*(x^2 - 4*x + 5)/(x - 1)^4: ser := series(gf, x, 42):
seq(coeff(ser, x, n), n = 0..40);

Mathematica

Table[n(n+2)(n+4), {n,0,40}] (* or *) CoefficientList[Series[3*x*(x^2 - 4*x + 5)/(x - 1)^4,{x,0,40}],x] (* James C. McMahon, Mar 05 2024 *)

Formula

a(n) = 8*Pochhammer(n/2, 3).

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

a(n) = 3 * A077415(n + 2).

From Klaus Purath, Aug 02 2024: (Start)

a(n)^2 = A028347(n+2)^3 + 4*A028347(n+2)^2.

a(n+1) - a(n) = A211441(n+2).

a(n) = 3*Sum_{i = 1..n} A028387(i). (End)

E.g.f.: exp(x)*x*(15 + 9*x + x^2). - Stefano Spezia, Aug 18 2024

From Amiram Eldar, Oct 03 2024: (Start)

Sum_{n>=1} 1/a(n) = 11/96.

Sum_{n>=1} (-1)^(n+1)/a(n) = 5/96. (End)