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%I #14 Aug 31 2020 11:15:20
%S 1,1,11,427,60719,32596531,68021747591,561032498484067,
%T 18426501525211985279,2417021848422676355879011,
%U 1267517634087900247917422974151,2658372329655374477213702898696297427,22300537841216964110498789350509161482874399
%N G.f.: Sum_{n>=0} (2^n + 1)^n * x^n / (1 + (2^n + 1)*x)^(n+1).
%C This sequence describes a specific case of the following identities:
%C (1) Sum_{n>=0} (q^n + p)^n * x^n / (1 + p*(q^n + p)*x)^(n+1) = Sum_{n>=0} (q^n - p)^n * x^n / (1 - p*(q^n - p)*x)^(n+1) ;
%C (2) Sum_{n>=0} (q^n + p)^n * exp(-p*(q^n + p)*x) * x^n/n! = Sum_{n>=0} (q^n - p)^n * exp( p*(q^n - p)*x) * x^n/n! ;
%C here, q = 2 and p = 1.
%F O.g.f.: Sum_{n>=0} (2^n + 1)^n * x^n / (1 + (2^n + 1)*x)^(n+1).
%F O.g.f.: Sum_{n>=0} (2^n - 1)^n * x^n / (1 - (2^n - 1)*x)^(n+1).
%F E.g.f.: Sum_{n>=0} (2^n + 1)^n * exp(-(2^n + 1)*x) * x^n/n!.
%F E.g.f.: Sum_{n>=0} (2^n - 1)^n * exp( (2^n - 1)*x) * x^n/n!.
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (2^k + 1)^n.
%F a(n) = Sum_{k=0..n} binomial(n,k) * (2^k - 1)^n.
%F a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Aug 31 2020
%e G.f.: A(x) = 1 + x + 11*x^2 + 427*x^3 + 60719*x^4 + 32596531*x^5 + 68021747591*x^6 + 561032498484067*x^7 + 18426501525211985279*x^8 + 2417021848422676355879011*x^9 + 1267517634087900247917422974151*x^10 + ...
%e where
%e A(x) = 1/(1 + 2*x) + 3*x/(1 + 3*x)^2 + 5^2*x^2/(1 + 5*x)^3 + 9^3*x^3/(1 + 9*x)^4 + 17^4*x^4/(1 + 17*x)^5 + 33^5*x^5/(1 + 33*x)^6 + ...
%e Also, by a series identity,
%e A(x) = 1 + x/(1 - x)^2 + 3^2*x^2/(1 - 3*x)^3 + 7^3*x^3/(1 - 7*x)^4 + 15^4*x^4/(1 - 15*x)^5 + 31^5*x^5/(1 - 31*x)^6 + ...
%e EXPONENTIAL GENERATING FUNCTION.
%e E.g.f.: B(x) = 1 + x + 11*x^2/2! + 427*x^3/3! + 60719*x^4/4! + 32596531*x^5/5! + 68021747591*x^6/6! + 561032498484067*x^7/7! + 18426501525211985279*x^8/8! + 2417021848422676355879011*x^9/9! + ...
%e where
%e B(x) = exp(-2*x) + 3*exp(-3*x)*x + 5^2*exp(-5*x)*x^2/2! + 9^3*exp(-9*x)*x^3/3! + 17^4*exp(-17*x)*x^4/4! + 33^5*exp(-33*x)*x^5/5! + ...
%e Also, by a series identity,
%e B(x) = 1 + 1*exp(1*x)*x + 3^2*exp(3*x)*x^2/2! + 7^3*exp(7*x)*x^3/3! + 15^4*exp(15*x)*x^4/4! + 31^5*exp(31*x)*x^5/5! + ...
%o (PARI) {a(n) = my(p=1,q=2,k=1);
%o A = sum(m=0,n, (q^m + p)^m * x^m / (1 + k*x + p*q^m*x +x*O(x^n))^(m+1) +x*O(x^n));
%o polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n) = my(p=1,q=2,k=1);
%o A = sum(m=0,n, (q^m - p)^m * x^m / (1 + k*x - p*q^m*x +x*O(x^n))^(m+1) +x*O(x^n));
%o polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n) = sum(k=0,n, (-1)^(n-k) * binomial(n,k) * (2^k + 1)^n )}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n) = sum(k=0,n, binomial(n,k) * (2^k - 1)^n )}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A337528.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 30 2020
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