cp's OEIS Frontend

A159478 a(n) = 2^(n^2+n) * C(1/2^n, n).

%I A159478 #16 Sep 08 2022 08:45:43
%S A159478 1,2,-6,140,-14570,6283452,-11049839724,78893138035608,
%T A159478 -2282580118745565210,267227101453296251927660,
%U A159478 -126415241162450125116966673796,241332381844862786094865482962203112,-1857025703922208959523779453799872508349700
%N A159478 a(n) = 2^(n^2+n) * C(1/2^n, n).
%C A159478 Sum_{n>=0} C(1/2^n, n) = 1.4306345243611686570661803375590... (A139823).
%H A159478 G. C. Greubel, <a href="/A159478/b159478.txt">Table of n, a(n) for n = 0..57</a>
%F A159478 G.f.: Sum_{n>=0} a(n)*x^n/2^(n^2+n) = Sum_{n>=0} log(1 + x/2^n)^n/n!.
%F A159478 a(n) = [x^n] (1 + 2^(n+1)*x)^(1/2^n).
%F A159478 a(n) ~ -(-1)^n * 2^(n^2)/n. - _Vaclav Kotesovec_, Jun 29 2018
%e A159478 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 A159478 A(x) = 1 + log(1+x/2) + log(1+x/4)^2/2! + log(1+x/8)^3/3! +...
%e A159478 Illustrate a(n) = [x^n] (1 + 2^(n+1)*x)^(1/2^n):
%e A159478 (1+4*x)^(1/2) = 1 + (2)*x - 2*x^2 + 4*x^3 - 10*x^4 +...
%e A159478 (1+8*x)^(1/4) = 1 + 2*x - (6)*x^2 + 28*x^3 - 154*x^4 +...
%e A159478 (1+16*x)^(1/8) = 1 + 2*x - 14*x^2 + (140)*x^3 - 1610*x^4 +...
%e A159478 (1+32*x)^(1/16) = 1 + 2*x - 30*x^2 + 620*x^3 - (14570)*x^4 +...
%e A159478 (1+64*x)^(1/32) = 1 + 2*x - 62*x^2 + 2604*x^3 - 123690*x^4 + (6283452)*x^5 +...
%t A159478 Table[2^(n^2 + n)*Binomial[1/2^n, n], {n, 0, 25}] (* _G. C. Greubel_, Jun 12 2018 *)
%o A159478 (PARI) a(n)=2^(n^2+n)*binomial(1/2^n,n)
%o A159478 (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 A159478 Cf. A139823.
%K A159478 sign
%O A159478 0,2
%A A159478 _Paul D. Hanna_, Apr 19 2009