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
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).
Programs
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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
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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);
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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 *)
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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)