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A134174 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(2^k,n).

%I A134174 #19 Feb 10 2024 04:45:58
%S A134174 1,1,4,44,1546,180096,69656776,90247564712,398349418563148,
%T A134174 6107204301234263072,331155704662551903747856,
%U A134174 64494012244416274972130226976,45679970430008422082570864807000804,118841746271330792971612314663523090972880,1144692720783363684402532942114790915481788335224
%N A134174 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(2^k,n).
%H A134174 G. C. Greubel, <a href="/A134174/b134174.txt">Table of n, a(n) for n = 0..59</a>
%F A134174 G.f.: Sum_{n>=0} log(1 + (2^n-1)*x)^n / n!.
%F A134174 a(n) = (1/n!) * Sum_{k=0..n} Stirling1(n,k) * (2^k-1)^n.
%F A134174 From _Vaclav Kotesovec_, Jul 02 2016: (Start)
%F A134174 a(n) ~ binomial(2^n,n).
%F A134174 a(n) ~ 2^(n^2) / n!.
%F A134174 a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).
%F A134174 (End)
%e A134174 G.f.: A(x) = 1 + x + 4*x^2 + 44*x^3 + 1546*x^4 + 180096*x^5 +...
%e A134174 where
%e A134174 A(x) = 1 + log(1+x) + log(1+3*x)^2/2! + log(1+7*x)^3/3! + log(1+15*x)^4/4! + log(1+31*x)^5/5! + log(1+63*x)^6/6! + log(1+127*x)^7/7! + log(1+255*x)^8/8! +...
%t A134174 Table[Sum[(-1)^(n-k)*Binomial[n,k]*Binomial[2^k,n],{k,0,n}],{n,0,15}] (* _Vaclav Kotesovec_, Jul 02 2016 *)
%o A134174 (PARI) {a(n)=local(A=1, X=x+x*O(x^n)); A=sum(k=0, n, log(1+(2^k-1)*X)^k/k!); polcoeff(A, n)}
%o A134174 for(n=0, 20, print1(a(n), ", "))
%o A134174 (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A134174 {a(n)=1/n!*sum(k=0,n,Stirling1(n,k)*(2^k-1)^n)}
%o A134174 for(n=0, 20, print1(a(n), ", "))
%Y A134174 Cf. A133990.
%K A134174 easy,nonn
%O A134174 0,3
%A A134174 _Paul D. Hanna_ and _Vladeta Jovovic_, Jan 13 2008