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A159318 a(n) = 2^(n^2+n) * binomial(2*n-1 + 1/2^n, n) / (n*2^n + 1).

%I A159318 #11 Sep 08 2022 08:45:43
%S A159318 1,2,26,1804,591894,860081340,5338683113364,138637536961147800,
%T A159318 14872932935424544987110,6538678365573711555851779180,
%U A159318 11717380780236748297970244719026812
%N A159318 a(n) = 2^(n^2+n) * binomial(2*n-1 + 1/2^n, n) / (n*2^n + 1).
%H A159318 G. C. Greubel, <a href="/A159318/b159318.txt">Table of n, a(n) for n = 0..56</a>
%F A159318 a(n) = [x^n] {(1-sqrt(1 - 2^(n+3)*x))/(2^(n+2)*x)}^(1/2^n).
%F A159318 G.f.: A(x) = Sum_{n>=0} a(n)*x^n/2^(n^2+n).
%F A159318 G.f.: A(x) = Sum_{n>=0} log(F(x/2^n))^n/n! where F(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function (A000108).
%F A159318 Radius of convergence of A(x) is |x| <= 1/2.
%F A159318 a(n) = [x^n] (1/(1 - 2^(n+1)*x)^(n + 1/2^n))/(n*2^n + 1). - _Paul D. Hanna_, Jun 15 2010
%e A159318 G.f.: A(x) = 1 + 2*x/2^2 + 26*x^2/2^6 + 1804*x^3/2^12 + 591894*x^4/2^20 + ...
%e A159318 G.f.: A(x) = Sum_{n>=0} log( 2^n*(1-sqrt(1 - 4*x/2^n))/(2*x) )^n/n!.
%e A159318 A(x) = 1 + log(F(x/2)) + log(F(x/4))^2/2! + log(F(x/8))^3/3! + ... where F(x) = (1-sqrt(1 - 4*x))/(2*x).
%e A159318 Special values.
%e A159318 A(1/2) = 1 + log(2) + log(4-4*sqrt(1/2))^2/2! + log(8-8*sqrt(3/4))^3/3! + log(16-16*sqrt(7/8))^4/4! + ...
%e A159318 A(1/2) = 1.70573970062357248928512380703308976974285275...
%e A159318 A(-1/2) = 1 + log(2*sqrt(2)-2) + log(4*sqrt(3/2)-4)^2/2! + log(8*sqrt(5/4)-8)^3/3! + log(16*sqrt(9/8)-16)^4/4! + ...
%e A159318 A(-1/2) = 0.81741280310249092844743171863299249334671633...
%e A159318 Illustrate a(n) = [x^n] {(1-sqrt(1-2^(n+3)*x))/(2^(n+2)*x)}^(1/2^n):
%e A159318 n=0: (1) + 2*x + 8*x^2 + 40*x^3 + 224*x^4 + 1344*x^5 + ...
%e A159318 n=1: 1 + (2)*x + 14*x^2 + 132*x^3 + 1430*x^4 + 16796*x^5 + ...
%e A159318 n=2: 1 + 2*x + (26)*x^2 + 476*x^3 + 10150*x^4 + 236060*x^5 + ...
%e A159318 n=3: 1 + 2*x + 50*x^2 + (1804)*x^3 + 76342*x^4 + 3534076*x^5 + ...
%e A159318 n=4: 1 + 2*x + 98*x^2 + 7020*x^3 + (591894)*x^4 + 54673468*x^5 + ...
%e A159318 n=5: 1 + 2*x + 194*x^2 + 27692*x^3 + 4660950*x^4 + (860081340)*x^5 + ...
%e A159318 coefficients in parenthesis form the initial terms of this sequence.
%t A159318 Table[2^(n^2 +n)*Binomial[2*n -1 +1/2^n, n]/(n*2^n +1), {n, 0, 50}] (* _G. C. Greubel_, Jun 26 2018 *)
%o A159318 (PARI) a(n)=2^(n^2+n)*binomial(2*n-1+1/2^n, n)/(n*2^n + 1)
%o A159318 (PARI) a(n)=polcoeff(((1-sqrt(1 - 2^(n+3)*x))/2^(n+2)/x)^(1/2^n),n)
%o A159318 (PARI) {a(n)=polcoeff(1/(1-2^(n+1)*x+x*O(x^n))^(n+1/2^n),n)/(n*2^n+1)} \\ _Paul D. Hanna_, Jun 15 2010
%o A159318 (Magma) [2^(n^2 +n)*Binomial(2*n -1 +1/2^n, n)/(n*2^n +1): n in [0..50]]; // _G. C. Greubel_, Jun 26 2018
%Y A159318 Cf. A159558, A159478, A158093, A000108.
%K A159318 nonn
%O A159318 0,2
%A A159318 _Paul D. Hanna_, Apr 22 2009