login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A159478 a(n) = 2^(n^2+n) * C(1/2^n, n). 5

%I #16 Sep 08 2022 08:45:43

%S 1,2,-6,140,-14570,6283452,-11049839724,78893138035608,

%T -2282580118745565210,267227101453296251927660,

%U -126415241162450125116966673796,241332381844862786094865482962203112,-1857025703922208959523779453799872508349700

%N a(n) = 2^(n^2+n) * C(1/2^n, n).

%C Sum_{n>=0} C(1/2^n, n) = 1.4306345243611686570661803375590... (A139823).

%H G. C. Greubel, <a href="/A159478/b159478.txt">Table of n, a(n) for n = 0..57</a>

%F G.f.: Sum_{n>=0} a(n)*x^n/2^(n^2+n) = Sum_{n>=0} log(1 + x/2^n)^n/n!.

%F a(n) = [x^n] (1 + 2^(n+1)*x)^(1/2^n).

%F a(n) ~ -(-1)^n * 2^(n^2)/n. - _Vaclav Kotesovec_, Jun 29 2018

%e G.f.: A(x) = 1 +2*x/2^2 -6*x^2/2^6 +140*x^3/2^12 -14570*x^4/2^20 +...

%e A(x) = 1 + log(1+x/2) + log(1+x/4)^2/2! + log(1+x/8)^3/3! +...

%e Illustrate a(n) = [x^n] (1 + 2^(n+1)*x)^(1/2^n):

%e (1+4*x)^(1/2) = 1 + (2)*x - 2*x^2 + 4*x^3 - 10*x^4 +...

%e (1+8*x)^(1/4) = 1 + 2*x - (6)*x^2 + 28*x^3 - 154*x^4 +...

%e (1+16*x)^(1/8) = 1 + 2*x - 14*x^2 + (140)*x^3 - 1610*x^4 +...

%e (1+32*x)^(1/16) = 1 + 2*x - 30*x^2 + 620*x^3 - (14570)*x^4 +...

%e (1+64*x)^(1/32) = 1 + 2*x - 62*x^2 + 2604*x^3 - 123690*x^4 + (6283452)*x^5 +...

%t Table[2^(n^2 + n)*Binomial[1/2^n, n], {n, 0, 25}] (* _G. C. Greubel_, Jun 12 2018 *)

%o (PARI) a(n)=2^(n^2+n)*binomial(1/2^n,n)

%o (Magma) SetDefaultRealField(RealField(250)); [Round(2^(n + n^2)*Gamma(1 + 1/2^n)/(Gamma(n+1)*Gamma(1 + 1/2^n - n))): n in [0..25]]; // _G. C. Greubel_, Jun 12 2018

%Y Cf. A139823.

%K sign

%O 0,2

%A _Paul D. Hanna_, Apr 19 2009

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)