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
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).
Programs
-
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);
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Mathematica
LinearRecurrence[ {4, 0, -8, -4}, {0, 1, 4, 14}, 28] -
PARI
concat(0, Vec(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.
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)