A331319 a(n) = [x^n](x - 2*x^3)/(1 - 2*x*(x + 1))^2.

0, 1, 4, 14, 48, 156, 496, 1544, 4736, 14352, 43072, 128224, 379136, 1114560, 3260160, 9494656, 27545600, 79642880, 229573632, 659951104, 1892478976, 5414755328, 15461117952, 44064835584, 125371383808, 356137570304, 1010187124736, 2861518086144, 8095486246912

Offset

0,3

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).

Formula

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

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

a(n) = n! [x^n] (1/9)*exp(x)*(sqrt(3)*(3*x+2)*sinh(sqrt(3)*x)+3*x*cosh(sqrt(3)*x)).

From Colin Barker, Jan 14 2020: (Start)

a(n) = ((1-sqrt(3))^n*(-2*sqrt(3) + 3*n) + (1+sqrt(3))^n*(2*sqrt(3) + 3*n)) / 18.

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

(End)

Maple

gf := (x - 2*x^3)/(1 - 2*x*(x + 1))^2: ser := series(gf, x, 32):
seq(coeff(ser, x, n), n=0..28);

Mathematica

LinearRecurrence[ {4, 0, -8, -4}, {0, 1, 4, 14}, 28]

Prog

(PARI) concat(0, Vec(x*(1 - 2*x^2) / (1 - 2*x - 2*x^2)^2 + O(x^30))) \\ Colin Barker, Jan 14 2020

Crossrefs

Cf. A322942 (Jacobsthal triangle), A331320, A331321.

Sequence in context: A022632 A027906 A047135 * A291254 A307127 A248957

Adjacent sequences: A331316 A331317 A331318 * A331320 A331321 A331322

Keyword

nonn,easy

Author

Peter Luschny, Jan 14 2020

Status

approved