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%I #20 Sep 08 2022 08:45:04
%S 1,8,288,23040,3225600,696729600,214592716800,89270570188800,
%T 48206107901952000,32780153373327360000,27404208220101672960000,
%U 27623441885862486343680000,33037636495491533667041280000,46252691093688147133857792000000
%N a(n) = (n+1)*2^n*(2*n)!.
%H Harry J. Smith, <a href="/A065141/b065141.txt">Table of n, a(n) for n = 0..100</a>
%F Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*1/8*exp(-1/(sqrt(2))*sqrt(x))*(x+sqrt(2)*sqrt(x))/x, x=0..infinity), n=0, 1...
%F Hypergeometric generating function, in Maple notation: exp(4*x)*(BesselI(0, 4*x)+4*x*BesselI(0, 4*x)+4*x*BesselI(1, 4*x)), i.e. a(0)=1 and a(n)= evalf(limit(n!^2*diff(exp(4*x)*(BesselI(0, 4*x)+4*x*BesselI(0, 4*x)+4*x*BesselI(1, 4*x)), x$n), x=0)), n=1, 2...
%t f[n_] := (n + 1)*2^n*(2*n)!; Array[f, 13, 0] (* _Robert G. Wilson v_, Jan 09 2018 *)
%o (PARI) { for (n=0, 100, write("b065141.txt", n, " ", (n + 1)*2^n*(2*n)!) ) } \\ _Harry J. Smith_, Oct 11 2009
%o (Magma) [(n+1)*2^n*Factorial(2*n): n in [0..20]]; // _Vincenzo Librandi_, Jan 13 2018
%K nonn
%O 0,2
%A _Karol A. Penson_, Oct 16 2001
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