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 A053482 Binomial transform of A029767. 3
 1, 4, 21, 142, 1201, 12336, 149989, 2113546, 33926337, 611660476, 12243073621, 269456124774, 6468249055921, 168191402251432, 4709596238204901, 141291441773619106, 4521383010795364609, 153727989225714801396, 5534225015581836134677 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 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 MATHEMATICA CoefficientList[Series[E^x/(1-3*x+2*x^2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *) CROSSREFS Sequence in context: A087761 A245503 A120368 * A158577 A006879 A228063 Adjacent sequences:  A053479 A053480 A053481 * A053483 A053484 A053485 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 15 2000 STATUS approved

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Last modified April 23 13:25 EDT 2021. Contains 343204 sequences. (Running on oeis4.)