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A053482 Binomial transform of A029767.

%I A053482 #24 Mar 02 2016 16:06:02
%S A053482 1,4,21,142,1201,12336,149989,2113546,33926337,611660476,12243073621,
%T A053482 269456124774,6468249055921,168191402251432,4709596238204901,
%U A053482 141291441773619106,4521383010795364609,153727989225714801396,5534225015581836134677
%N A053482 Binomial transform of A029767.
%C A053482 This is the column k=3 of an array T(n,k) = A181783(n,k) defined by T(n,0)=T(0,k)=1 and T(n,k) = n*(k-1)*T(n-1,k) +T(n,k-1), which starts
%C A053482    1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,...
%C A053482    1,   1,   2,   4,   7,  11,  16,  22,  29,  37,  46,...
%C A053482    1,   1,   5,  21,  63, 151, 311, 575, 981,1573,2401,...
%C A053482    1,   1,  16, 142, 709,2521,7186,17536,38137,75889,140716,...
%C A053482    1,   1,  65,1201,9709,50045,193765,614629,1682465,4110913,9176689,...
%C A053482 Column k=2 is A000522. The e.g.f. for column k is E_k(z) = E_(k-1)(z)/[1-(k-1)] = exp(z)/prod_{j=1..k-1} (1-j*z). - _Richard Choulet_, Dec 17 2012
%H A053482 Vincenzo Librandi, <a href="/A053482/b053482.txt">Table of n, a(n) for n = 0..200</a>
%F A053482 E.g.f.: exp(x)*(2/(1-2x)-1/(1-x))=exp(x)/(1-3x+2x^2); a(n)=sum{k=0..n, C(n,k)*k!*(2^(k+1)-1)}; a(n)=n!*sum{k=0..n, (2^(n-k+1)-1)/k!}; a(n)=int(x^n*(exp((1-x)/2)-exp(1-x)),x,1,infty); a(n)=2*A010844(n)-A000522(n); - _Paul Barry_, Jan 28 2008
%F A053482 Conjecture: a(n) -(3*n+1)*a(n-1) +(2*n+3)*(n-1)*a(n-2) -2*(n-1)*(n-2)*a(n-3)=0. - _R. J. Mathar_, Sep 29 2012
%F A053482 a(n) = 3*n*a(n-2)-2*n*(n-1)*a(n-2)+1, derived from the array defined in the comment, which proves the previous conjecture. - _Richard Choulet_, Dec 17 2012
%F A053482 a(n) ~ n! * 2^(n+1)*exp(1/2). - _Vaclav Kotesovec_, Oct 02 2013
%t A053482 CoefficientList[Series[E^x/(1-3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 02 2013 *)
%K A053482 nonn
%O A053482 0,2
%A A053482 _N. J. A. Sloane_, Jan 15 2000