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%I #9 Nov 16 2022 05:06:01
%S 1,3,13,73,527,4775,52589,683785,10257031,174370039,3313031765,
%T 69573669113,1600194393695,40004859850567,1080131215981693,
%U 31323805263501865,971037963168623351,32044252784564701655,1121548847459764820069,41497307356011298866841,1618394986884440656855375
%N a(n) = (2*n - 1)*a(n - 1) + 2^n for n >= 1, a(0) = 1.
%C A weighted sum of quotients of double factorials.
%C a(n) are the row sum of triangle A126063.
%H P. Luschny, <a href="http://www.luschny.de/math/seq/variations.html">Variants of Variations</a>.
%F a(n) = (2n)!/(n! 2^n) Sum(k=0..n, 4^k k!/(2k)!)
%F a(n) = 2^n Gamma(n+1/2) Sum(k=0..n, 1/Gamma(k+1/2))
%F a(n) = Sum(k=0..n, 2^k n!!/k!!) [n!! defined as A001147(n), Gottfried Helms]
%F a(n) = Sum(k=0..n, 2^(2k-n)((n+1)! Catalan(n))/((k+1)! Catalan(k))) [Catalan(n) A000108]
%F a(n) = Sum(k=0..n, 2^(2k-n) QuadFact(n)/QuadFact(k)) [QuadFact(n) A001813]
%F a(n) = Sum(k=0..n, 2^(2k-n) (-1)^(n-k) A097388(n)/A097388(k) )
%F a(n) = A001147(n) Sum(k=0..n, 2^k / A001147(k))
%F a(n) = A128195(n)/A005408(n)
%F a(n) = A128195(n-1)+A000079(n) (if n>0)
%F Recursive form: a(n) = (2n-1)*a(n-1) + 2^n; a(0) = 1 [Gottfried Helms]
%F Note: The following constants will be used in the next formulas.
%F K = (1-exp(1)*Gamma(1/2,1))/Gamma(1/2)
%F M = sqrt(2)(1+exp(1)(Gamma(1/2)-Gamma(1/2,1)))
%F Generalized form: For x>0
%F a(x) = 2^x(exp(1)*Gamma(x+1/2,1) + K*Gamma(x+1/2))
%F Asymptotic formula:
%F a(n) ~ 2^n*(1+(exp(1)+K)*(n-1/2)!)
%F a(n) ~ M(2exp(-1)(n-1/(24*n+19/10*1/n)))^n
%p a := n -> `if`(n=0,1,(2*n-1)*a(n-1)+2^n);
%t a[n_] := Sum[2^k*((2*n-1)!!/(2*k-1)!!), {k, 0, n}]; Table[a[n], {n, 0, 14}] (* _Jean-François Alcover_, Jun 28 2013 *)
%Y Cf. A128195, A001147, A126063.
%K easy,nonn
%O 0,2
%A _Peter Luschny_, Feb 26 2007
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