A144704 a(n) = 4^n*(1-2*n).

1, -4, -48, -320, -1792, -9216, -45056, -212992, -983040, -4456448, -19922944, -88080384, -385875968, -1677721600, -7247757312, -31138512896, -133143986176, -566935683072, -2405181685760, -10170482556928

Offset: 0

Author

Paul Barry, Sep 19 2008

Keywords

• easy
• sign

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (8,-16).

Crossrefs

Hankel transform of A100320.

Cf. A000302, A165747.

Programs

Mathematica

LinearRecurrence[{8,-16},{1,-4},30] (* Harvey P. Dale, Jun 12 2019 *)

SageMath

[4^n*(1-2*n) for n in (0..30)] # G. C. Greubel, Jun 16 2022

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