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A144704
a(n) = 4^n*(1-2*n).
5
1, -4, -48, -320, -1792, -9216, -45056, -212992, -983040, -4456448, -19922944, -88080384, -385875968, -1677721600, -7247757312, -31138512896, -133143986176, -566935683072, -2405181685760, -10170482556928
OFFSET
0,2
COMMENTS
With the n-th term of A000984 (C(2n,n)) as numerator, |a(n)| is the denominator of the probability that a random walk with steps of +-1 will return to the starting point for the first time after 2n steps. - Shel Kaphan, Jan 12 2023
FORMULA
G.f.: (1-12*x)/(1-4*x)^2.
From Amiram Eldar, Aug 05 2020: (Start)
Sum_{n>=0} 1/a(n) = 1 - arctanh(1/2)/2.
Sum_{n>=0} (-1)^(n+1)/a(n) = 1 + arctan(1/2)/2. (End)
E.g.f.: (1 - 8*x)*exp(4*x). - G. C. Greubel, Jun 16 2022
Sum_{n >= 1} x^(2*n-1)/a(n) = 1/4 * log((1 - x/2)/(1 + x/2)). Eldar's two summations above follow from this on setting x = 1 and x = i. - Peter Bala, Jul 08 2024
MATHEMATICA
LinearRecurrence[{8, -16}, {1, -4}, 30] (* Harvey P. Dale, Jun 12 2019 *)
PROG
(SageMath) [4^n*(1-2*n) for n in (0..30)] # G. C. Greubel, Jun 16 2022
CROSSREFS
Hankel transform of A100320.
Sequence in context: A217153 A129002 A291623 * A091904 A192831 A158681
KEYWORD
easy,sign
AUTHOR
Paul Barry, Sep 19 2008
STATUS
approved