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%I #36 Aug 10 2018 12:02:33
%S 1,1,6,58,798,14150,307076,7881756,233536532,7844786248,294582696686,
%T 12228351266210,556017625969246,27482790417322218,1467194712330407238,
%U 84134395928742550138,5157545958316518485420,336574587493456290969620,23296320082405927961459550,1704662916739625989249415610,131480805016085834305348796128
%N G.f.: Sum_{n>=0} (1 + (1+x)^n)^n / (2 + (1+x)^n)^(n+1).
%C The following identity holds for |y| <= 1 and fixed real k > 0:
%C Sum_{n>=0} (k + y^n)^n/(1+k + y^n)^(n+1) = Sum_{n>=0} (y^n - 1)^n/(1+k - k*y^n)^(n+1).
%H Vaclav Kotesovec, <a href="/A302598/b302598.txt">Table of n, a(n) for n = 0..360</a> (terms 0..200 from Paul D. Hanna)
%F G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (2 - (1+x)^n)^(n+1).
%F G.f.: Sum_{n>=0} ((1+x)^n + 1)^n / (2 + (1+x)^n)^(n+1).
%F a(n) ~ c * d^n * n! / sqrt(n), where d = A317904 = 3.9561842030261697545408021818783008332999988095... and c = 0.274656497660429769528095546948772676444158... - _Vaclav Kotesovec_, Aug 09 2018
%e G.f.: A(x) = 1 + x + 6*x^2 + 58*x^3 + 798*x^4 + 14150*x^5 + 307076*x^6 + 7881756*x^7 + 233536532*x^8 + 7844786248*x^9 + 294582696686*x^10 + ...
%e such that
%e A(x) = 1/3 + (1 + (1+x))/(2 + (1+x))^2 + (1 + (1+x)^2)^2/(2 + (1+x)^2)^3 + (1 + (1+x)^3)^3/(2 + (1+x)^3)^4 + (1 + (1+x)^4)^4/(2 + (1+x)^4)^5 + (1 + (1+x)^5)^5/(2 + (1+x)^5)^6 + (1 + (1+x)^6)^6/(2 + (1+x)^6)^7 + ...
%e Also,
%e A(x) = 1 + ((1+x) - 1)/(2 - (1+x))^2 + ((1+x)^2 - 1)^2/(2 - (1+x)^2)^3 + ((1+x)^3 - 1)^3/(2 - (1+x)^3)^4 + ((1+x)^4 - 1)^4/(2 - (1+x)^4)^5 + ((1+x)^5 - 1)^5/(2 - (1+x)^5)^6 + ((1+x)^6 - 1)^6/(2 - (1+x)^6)^7 + ...
%e RELATED INFINITE SERIES.
%e (1) At x = -1/2: the following sums are equal
%e S1 = Sum_{n>=0} 2^n * (2^n + 1)^n/(2^(n+1) + 1)^(n+1)
%e S1 = Sum_{n>=0} (-2)^n * (2^n - 1)^n/(2^(n+1) - 1)^(n+1).
%e Explicitly,
%e S1 = 1/3 + 2*3/5^2 + 4*5^2/9^3 + 8*9^3/17^4 + 16*17^4/33^5 + 32*33^5/65^6 + 64*65^6/129^7 + 128*129^7/257^8 + 256*257^8/513^9 + 512*513^9/1025^10 + ...
%e S1 = 1 - 2*1/3^2 + 4*3^2/7^3 - 8*7^3/15^4 + 16*15^4/31^5 - 32*31^5/63^6 + 64*63^6/127^7 - 128*127^7/255^8 + 256*255^8/511^9 - 512*511^9/1023^10 + ...
%e where S1 = 0.84714730053329880291591114748812485885366310294051236295420...
%e (2) At x = -2/3: the following sums are equal
%e S2 = Sum_{n>=0} 3^n * (1 + 3^n)^n / (2*3^n + 1)^(n+1)
%e S2 = Sum_{n>=0} (-3)^n * (3^n - 1)^n / (2*3^n - 1)^(n+1).
%e Explicitly,
%e S2 = 1/3 + 3*4/7^2 + 9*10^2/19^3 + 27*28^3/55^4 + 81*82^4/163^5 + 243*244^5/487^6 + 729*730^6/1459^7 + 2187*2188^7/4375^8 + 6561*6562^8/13123^9 + 19683*19684^9/39367^10 + ...
%e S2 = 1 - 3*2^1/5^2 + 9*8^2/17^3 - 27*26^3/53^4 + 81*80^4/161^5 - 243*242^5/485^6 + 729*728^6/1457^7 - 2187*2186^7/4373^8 + 6561*6560^8/13121^9 - 19683*19682^9/39365^10 + ...
%e where S2 = 0.837457334418049175936255584889342515316005199043439291643371...
%e (3) At x = -1/3: the following sums are equal
%e S3 = Sum_{n>=0} 3^n * (2^n + 3^n)^n/(2*3^n + 2^n)^(n+1)
%e S3 = Sum_{n>=0} (-3)^n * (3^n - 2^n)^n/(2*3^n - 2^n)^(n+1).
%e Explicitly,
%e S3 = 1/3 + 3*5/8^2 + 9*13^2/22^3 + 27*35^3/62^4 + 81*97^4/178^5 + 243*275^5/518^6 + 729*793^6/1522^7 + 2187*2315^7/4502^8 + 6561*6817^8/13378^9 + 19683*20195^9/39878^10 + ...
%e S3 = 1 - 3*1/4^2 + 9*5^2/14^3 - 27*19^3/46^4 + 81*65^4/146^5 - 243*211^5/454^6 + 729*665^6/1394^7 - 2187*2059^7/4246^8 + 6561*6305^8/12866^9 - 19683*19171^9/38854^10 + ...
%e where S3 = 0.867357695200699139470956415922046910279987551651352471994920...
%o (PARI) {a(n) = my(A=1,o=x*O(x^n)); A = sum(m=0,n,((1+x +o)^m - 1)^m / (2 - (1+x +o)^m)^(m+1)); polcoeff(A,n)}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A302700, A122400, A302614, A302615.
%Y Cf. A317662, A317663, A317664.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Apr 10 2018
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