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A067273 a(n) = n*(a(n-1)*2+1), a(0) = 0.

%I A067273 #37 Jun 15 2025 12:40:18
%S A067273 0,1,6,39,316,3165,37986,531811,8508984,153161721,3063234430,
%T A067273 67391157471,1617387779316,42052082262229,1177458303342426,
%U A067273 35323749100272795,1130359971208729456,38432239021096801521,1383560604759484854774,52575302980860424481431,2103012119234416979257260,88326509007845513128804941
%N A067273 a(n) = n*(a(n-1)*2+1), a(0) = 0.
%H A067273 Harvey P. Dale, <a href="/A067273/b067273.txt">Table of n, a(n) for n = 0..403</a>
%F A067273 E.g.f.: x*exp(x)/(1-2*x). a(n) = n!*Sum_{k=1..n} 2^(k-1)/(n-k)! = n*A010844(n-1). - _Vladeta Jovovic_, Feb 09 2003
%F A067273 a(n) ~ n! * exp(1/2) * 2^(n-1). - _Vaclav Kotesovec_, Oct 05 2013
%F A067273 a(n) = n*hypergeom([1,1-n], [], -2). - _Peter Luschny_, May 09 2017
%F A067273 a(n) = Sum_{k=1..n} 2^(k-1)*k!*binomial(n,k). - _Ridouane Oudra_, Jun 15 2025
%p A067273 a := n -> n*hypergeom([1,1-n],[],-2):
%p A067273 seq(simplify(a(n)), n=0..17); # _Peter Luschny_, May 09 2017
%t A067273 FoldList[2 #1*#2 + #2 &, 0, Range[19]] (* _Robert G. Wilson v_, Jul 07 2012 *)
%t A067273 a[n_] := 2^(n-1)*Sqrt[E]*n*Gamma[n,1/2];
%t A067273 Table[a[n] // FullSimplify, {n,0,20}] (* _Gerry Martens_, Jun 28 2015 *)
%t A067273 nxt[{n_,a_}]:={n+1,(n+1)(2*a+1)}; NestList[nxt,{0,0},20][[;;,2]] (* _Harvey P. Dale_, Jun 26 2023 *)
%Y A067273 Cf. A007526.
%K A067273 nonn
%O A067273 0,3
%A A067273 _Reinhard Zumkeller_, Feb 21 2002
%E A067273 More terms from _Vladimir Joseph Stephan Orlovsky_, Nov 15 2008