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A331320
a(n) = [x^n] ((x + 1)*(2*x - 1)*(2*x^2 - 1))/(2*x^2 + 2*x - 1)^2.
3
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
0,2
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)
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);
PROG
(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
CROSSREFS
Cf. A322942 (Jacobsthal triangle), A331319, A331321.
Sequence in context: A148801 A131910 A205775 * A148802 A255712 A194690
KEYWORD
nonn,easy
AUTHOR
Peter Luschny, Jan 14 2020
STATUS
approved