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A133990 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(2^k + n - 1,n). 3

%I #24 Feb 11 2017 02:16:09

%S 1,1,5,71,2747,306861,106709627,123122238887,492425723170553,

%T 7012142056418141897,361269845371107759765065,

%U 68033187103968192731087467135,47171609221094330538117045468744655

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

%H G. C. Greubel, <a href="/A133990/b133990.txt">Table of n, a(n) for n = 0..59</a>

%F G.f.: Sum_{n>=0} (-log(1 - (2^n-1)*x))^n / n!.

%F a(n) = (1/n!) * Sum_{k=0..n} (-1)^(n-k) * Stirling1(n,k) * (2^k-1)^n.

%F From _Vaclav Kotesovec_, Jul 02 2016: (Start)

%F a(n) ~ binomial(2^n,n).

%F a(n) ~ 2^(n^2) / n!.

%F a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)).

%F (End)

%e G.f.: A(x) = 1 + x + 5*x^2 + 71*x^3 + 2747*x^4 + 306861*x^5 +...

%e where

%e 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! +...

%p A133990 := proc(n) add((-1)^(n-k)*binomial(n,k)*binomial(2^k+n-1,n),k=0..n) ; end: seq(A133990(n),n=0..15) ; # _R. J. Mathar_, Jan 30 2008

%t Table[Sum[(-1)^(n-k) Binomial[n,k]Binomial[2^k+n-1,n],{k,0,n}], {n,0,15}] (* _Harvey P. Dale_, Nov 24 2011 *)

%o (PARI) {a(n)=local(A=1,X=x+x*O(x^n)); A=sum(k=0,n,log(1/(1-(2^k-1)*X))^k/k!); polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

%o {a(n)=1/n!*sum(k=0,n,(-1)^(n-k)*Stirling1(n,k)*(2^k-1)^n)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A134174.

%K easy,nonn

%O 0,3

%A _Paul D. Hanna_ and _Vladeta Jovovic_, Jan 21 2008

%E More terms from _R. J. Mathar_, Jan 30 2008

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Last modified March 28 04:58 EDT 2024. Contains 371235 sequences. (Running on oeis4.)