A053482 Binomial transform of A029767.

1, 4, 21, 142, 1201, 12336, 149989, 2113546, 33926337, 611660476, 12243073621, 269456124774, 6468249055921, 168191402251432, 4709596238204901, 141291441773619106, 4521383010795364609, 153727989225714801396, 5534225015581836134677

Offset: 0

Author

N. J. A. Sloane, Jan 15 2000

Keywords

• nonn

Comments

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

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,...

1, 1, 2, 4, 7, 11, 16, 22, 29, 37, 46,...

1, 1, 5, 21, 63, 151, 311, 575, 981,1573,2401,...

1, 1, 16, 142, 709,2521,7186,17536,38137,75889,140716,...

1, 1, 65,1201,9709,50045,193765,614629,1682465,4110913,9176689,...

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Programs

Mathematica

CoefficientList[Series[E^x/(1-3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

Formula

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

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

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

a(n) ~ n! * 2^(n+1)*exp(1/2). - Vaclav Kotesovec, Oct 02 2013