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G.f.: Sum_{n>=0} (3 + (1+x)^n)^n / (4 + (1+x)^n)^(n+1).
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%I #12 Aug 09 2018 12:05:34

%S 1,1,10,130,2390,56714,1644138,56327820,2226708772,99761490536,

%T 4995375316146,276464859358474,16757956600528786,1104116777798713154,

%U 78565751676021256606,6004629888868350015506,490572645247461234631946,42665124626946741636482996,3935474733572880332326074450,383756013888633346483785849474

%N G.f.: Sum_{n>=0} (3 + (1+x)^n)^n / (4 + (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 Paul D. Hanna, <a href="/A302615/b302615.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: Sum_{n>=0} ((1+x)^n - 1)^n / (4 - 3*(1+x)^n)^(n+1).

%F a(n) ~ c * d^n * n! / sqrt(n), where d = 5.2709551504518355656831902094014170087... and c = 0.26621450180820822374221893929... - _Vaclav Kotesovec_, Aug 09 2018

%e G.f.: A(x) = 1 + x + 10*x^2 + 130*x^3 + 2390*x^4 + 56714*x^5 + 1644138*x^6 + 56327820*x^7 + 2226708772*x^8 + 99761490536*x^9 + ...

%e such that

%e A(x) = 1/5 + (3 + (1+x))/(4 + (1+x))^2 + (3 + (1+x)^2)^2/(4 + (1+x)^2)^3 + (3 + (1+x)^3)^3/(4 + (1+x)^3)^4 + (3 + (1+x)^4)^4/(4 + (1+x)^4)^5 + (3 + (1+x)^5)^5/(4 + (1+x)^5)^6 + (3 + (1+x)^6)^6/(4 + (1+x)^6)^7 + ...

%e Also,

%e A(x) = 1 + ((1+x) - 1)/(4 - 3*(1+x))^2 + ((1+x)^2 - 1)^2/(4 - 3*(1+x)^2)^3 + ((1+x)^3 - 1)^3/(4 - 3*(1+x)^3)^4 + ((1+x)^4 - 1)^4/(4 - 3*(1+x)^4)^5 + ((1+x)^5 - 1)^5/(4 - 3*(1+x)^5)^6 + ((1+x)^6 - 1)^6/(4 - 3*(1+x)^6)^7 + ...

%o (PARI) {a(n) = my(A=1, o=x*O(x^n)); A = sum(m=0, n, ((1+x +o)^m - 1)^m / (4 - 3*(1+x +o)^m)^(m+1)); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A122400, A302598, A302614.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 10 2018