A328000 a(n) = Sum_{k=0..n}(k!*(n - k)!)/(floor(k/2)!*floor((n - k)/2)!)^2.

1, 2, 5, 16, 28, 96, 160, 512, 896, 2560, 4864, 12288, 25600, 57344, 131072, 262144, 655360, 1179648, 3211264, 5242880, 15466496, 23068672, 73400320, 100663296, 343932928, 436207616, 1593835520, 1879048192, 7314866176, 8053063680, 33285996544, 34359738368

Offset: 0

Author

Peter Luschny, Oct 01 2019

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,12,0,-48,0,64).

Crossrefs

Cf. A327999, A328001.

Cf. A056040, A002699, A001477, A081908, A145018.

Programs

Magma

[IsOdd(n) select 2^(n - 1)*(n + 1) else 2^(n - 5)*(n*(n + 2) + 32):n in [0..30]]; // Marius A. Burtea, Feb 05 2020

Maple

swing := n -> n!/iquo(n,2)!^2: a := n -> add(swing(k)*swing(n-k), k=0..n):
seq(`if`(irem(n, 2) = 0, 2 + n*(n + 2)/16, n + 1)*2^(n - 1), n=0..31);

Mathematica

A328000List[len_] := CoefficientList[Series[(4 x^2 - x - 1)^2 / (1 - 4 x^2)^3 , {x, 0, len}], x]; A328000List[31]
LinearRecurrence[{0,12,0,-48,0,64},{1,2,5,16,28,96},40] (* Harvey P. Dale, Jun 19 2022 *)

PARI

x='x + O('x^32);
Vec(serlaplace(((3*x + 8)*sinh(2*x) + (2*x^2 + 16*(x + 1))*cosh(2*x))/16))

PARI

Vec((1 + x - 4*x^2)^2 / ((1 - 2*x)^3*(1 + 2*x)^3) + O(x^30)) \\ Colin Barker, Feb 05 2020

Formula

a(n) = Sum_{k=0..n} s(k)*s(n-k) where s(n) = A056040(n).

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

a(n) = 2^(n - 5)*(n*(n + 2) + 32) if n even else 2^(n - 1)*(n + 1).

a(2*n) = A327999(n).

a(2*n-1) = A002699(n), (with a(-1) = 0).

a(2^n-1) = 2^(2^n - 2 + n) for n >= 1.

2*a(2*n)/2^n = A081908(n+1).

4*a(2*n)/4^n = A145018(n+1).

2*a(2*n-1)/4^n = A001477(n).

From Stefano Spezia, Oct 19 2019: (Start)

a(n) = n! [x^n] (1/32)*exp(-2*x)*(8 + exp(4*x)*(8 + x)*(3 + 2*x) + x*(13 + 2*x)).

a(n) = 12*a(n-2) - 48*a(n-4) + 64*a(n-6) for n > 5. (End)