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%I #19 Mar 13 2019 05:25:17
%S 1,2,14,464,62498,32904512,68218246274,561509337384704,
%T 18430991691638470658,2417187707139291150820352,
%U 1267541805135644381508133312514,2658386272482288686337601015526764544,22300569741768576670596245546950890721591298,748287098859335987804516114266959555239667124355072,100433559671950042126701317490712783944232737693284609851394
%N G.f.: Sum_{n>=0} (2^n + 1)^n * x^n / (1 + 2^n*x)^(n+1).
%H Paul D. Hanna, <a href="/A324306/b324306.txt">Table of n, a(n) for n = 0..75</a>
%F G.f.: Sum_{n>=0} (2^n + 1)^n * x^n / (1 + 2^n*x)^(n+1).
%F G.f.: Sum_{n>=0} (2^n - 1)^n * x^n / (1 - 2^n*x)^(n+1).
%F a(n) = Sum_{k=0..n} binomial(n,k) * (2^n - 2^k)^(n-k).
%F a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * (2^n + 2^k)^(n-k).
%F a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Mar 13 2019
%e G.f.: A(x) = 1 + 2*x + 14*x^2 + 464*x^3 + 62498*x^4 + 32904512*x^5 + 68218246274*x^6 + 561509337384704*x^7 + 18430991691638470658*x^8 + ...
%e such that
%e A(x) = 1/(1+x) + (2 + 1)*x/(1 + 2*x)^2 + (2^2 + 1)^2*x^2/(1 + 2^2*x)^3 + (2^3 + 1)^3*x^3/(1 + 2^3*x)^4 + (2^4 + 1)^4*x^4/(1 + 2^4*x)^5 + (2^5 + 1)^5*x^5/(1 + 2^5*x)^6 + (2^6 + 1)^6*x^6/(1 + 2^6*x)^7 + ...
%e Also,
%e A(x) = 1/(1-x) + (2 - 1)*x/(1 - 2*x)^2 + (2^2 - 1)^2*x^2/(1 - 2^2*x)^3 + (2^3 - 1)^3*x^3/(1 - 2^3*x)^4 + (2^4 - 1)^4*x^4/(1 - 2^4*x)^5 + (2^5 - 1)^5*x^5/(1 - 2^5*x)^6 + (2^6 - 1)^6*x^6/(1 - 2^6*x)^7 + ...
%e Explicitly,
%e A(x) = 1/(1+x) + 3*x/(1 + 2*x)^2 + 5^2*x^2/(1 + 2^2*x)^3 + 9^3*x^3/(1 + 2^3*x)^4 + 17^4*x^4/(1 + 2^4*x)^5 + 33^5*x^5/(1 + 2^5*x)^6 + 65^6*x^6/(1 + 2^6*x)^7 + ...
%e and
%e A(x) = 1/(1-x) + x/(1 - 2*x)^2 + 3^2*x^2/(1 - 2^2*x)^3 + 7^3*x^3/(1 - 2^3*x)^4 + 15^4*x^4/(1 - 2^4*x)^5 + 31^5*x^5/(1 - 2^5*x)^6 + 63^6*x^6/(1 - 2^6*x)^7 + ...
%t Table[1 + Sum[Binomial[n, k]*(2^n - 2^k)^(n - k), {k, 0, n-1}], {n, 0, 15}] (* _Vaclav Kotesovec_, Mar 13 2019 *)
%o (PARI) {a(n) = my(A = sum(m=0, n+1, (2^m + 1)^m*x^m/(1 + 2^m*x +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = my(A = sum(m=0, n+1, (2^m - 1)^m*x^m/(1 - 2^m*x +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = sum(k=0,n, binomial(n,k) * (2^n - 2^k)^(n-k) )}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n) = sum(k=0,n, (-1)^k * binomial(n,k) * (2^n + 2^k)^(n-k) )}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A324308, A324307.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 09 2019
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