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 A224733 a(n) = binomial(2*n,n)^n. 3

%I

%S 1,2,36,8000,24010000,1016255020032,622345892187672576,

%T 5608296349498479967469568,752711194884611945703392100000000,

%U 1518219588672387021538193329290752000000000,46343145866349732399475841723454160331675252923826176

%N a(n) = binomial(2*n,n)^n.

%C a(n) = A000984(n)^n, where A000984 is the central binomial coefficients.

%F Logarithmic derivative of A224732 (when ignoring initial term a(0)=1).

%F a(n) ~ exp(-1/8) * 4^(n^2) / (n^(n/2) * Pi^(n/2)). - _Vaclav Kotesovec_, Mar 04 2014

%e L.g.f.: L(x) = 2*x + 36*x^2/2 + 8000*x^3/3 + 24010000*x^4/4 + 1016255020032*x^5/5 +...

%e Equivalently,

%e L(x) = 2*x + 6^2*x^2/2 + 20^3*x^3/3 + 70^4*x^4/4 + 252^5*x^5/5 + 924^6*x^6/6 + 3432^7*x^7/7 + 12870^8*x^8/8 +...+ A000984(n)^n*x^n/n +...

%e where exponentiation yields an integer series:

%e exp(L(x)) = 1 + 2*x + 20*x^2 + 2704*x^3 + 6008032*x^4 + 203263062688*x^5 + 103724721990326528*x^6 +...+ A224732(n)*x^n +...

%t Table[Binomial[2n,n]^n,{n,0,10}] (* _Harvey P. Dale_, Apr 19 2016 *)

%o (PARI) {a(n)=binomial(2*n,n)^n}

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

%Y Cf. A224732, A000984.

%K nonn,nice

%O 0,2

%A _Paul D. Hanna_, Apr 16 2013

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Last modified March 27 12:09 EDT 2023. Contains 361570 sequences. (Running on oeis4.)