A331320 a(n) = [x^n] ((x + 1)*(2*x - 1)*(2*x^2 - 1))/(2*x^2 + 2*x - 1)^2.

1, 3, 8, 26, 80, 244, 736, 2200, 6528, 19248, 56448, 164768, 478976, 1387328, 4005376, 11530624, 33107968, 94839552, 271091712, 773380608, 2202374144, 6261404672, 17774206976, 50384312320, 142636515328, 403306786816, 1139055820800, 3213593911296, 9057375289344

Offset: 0

Author

Peter Luschny, Jan 14 2020

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,0,-8,-4).

Crossrefs

Cf. A322942 (Jacobsthal triangle), A331319, A331321.

Programs

Maple

a := proc(n) option remember; if n < 3 then return [1, 3, 8][n+1] fi;
(12*(n - 3)*a(n-3) + (14*n - 6)*a(n-2) + (70 - 4*n)*a(n-1))/(n+19) end:
seq(a(n), n=0..28);
# Alternative:
gf := ((x + 1)*(2*x - 1)*(2*x^2 - 1))/(2*x^2 + 2*x - 1)^2:
ser := series(gf, x, 32): seq(coeff(ser, x, n), n=0..28);

Mathematica

LinearRecurrence[{4,0,-8,-4},{1,3,8,26,80},40] (* Harvey P. Dale, Jun 14 2025 *)

PARI

Vec((1 + x)*(1 - 2*x)*(1 - 2*x^2) / (1 - 2*x - 2*x^2)^2 + O(x^30)) \\ Colin Barker, Jan 14 2020

Formula

a(n) = Sum_{k=0..n} A322942(n,k)*(k+1).

a(n) = (12*(n - 3)*a(n-3) + (14*n - 6)*a(n-2) + (70 - 4*n)*a(n-1))/(n + 19).

Let h(k) = (1+k)*exp((1+k)*x)*(3*x+12-4*k)/18 then

a(n) = n!*[x^n](h(sqrt(3)) + h(-sqrt(3)) + 1).

From Colin Barker, Jan 14 2020: (Start)

a(n) = 4*a(n-1) - 8*a(n-3) - 4*a(n-4) for n>4.

a(n) = (-8*sqrt(3)*((1-sqrt(3))^n - (1+sqrt(3))^n) + 3*((1-sqrt(3))^n + (1+sqrt(3))^n)*n) / 18 for n>0.

(End)