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%I #20 Sep 17 2015 03:03:57
%S 1,3,10,53,286,2227,16682,164089,1619506,19139231,228289282,
%T 3233324437,45177961154,727739459371,11876754782146,215081192780657,
%U 3943187021584642,80187760712122039,1635670770628012706
%N E.g.f.: Sum_{k>=1} (x^k/k! / Product_{i=1..k} (1-x^i)).
%H Vaclav Kotesovec, <a href="/A133148/b133148.txt">Table of n, a(n) for n = 1..400</a>
%F E.g.f.: let B(x)=(1 - G(0))/(1-x) where G(k) = 1 - 1/(1-x^(k+1))/(k+1)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction); then the e.g.f. equals d/dx B(x). - _Sergei N. Gladkovskii_, Jan 22 2013
%p A133148 := proc(n) local egf,k,i,tmp; egf := 0 ; for k from 1 to n do tmp := x^k/k! ; for i from 1 to k do tmp := taylor( tmp/(1-x^i),x=0,n+1) ; od: egf := egf+tmp ; od: coeftayl(egf*n!,x=0,n) ; end: seq(A133148(n),n=1..25) ; # _R. J. Mathar_, Sep 28 2007
%t Rest[ Range[0, 15]!* CoefficientList[ Series[ Sum[x^k/k!/Product[(1 - x^i), {i, k}], {k, 15}], {x, 0, 15}], x]] (* _Robert G. Wilson v_, Oct 07 2007 *)
%t max = 18; Clear[g]; g[max + 1] = 1; g[k_] := g[k] = 1 - 1/(1 - x^(k + 1))/(k + 1)/(1 - x/(x - 1/g[k + 1])); b[x_] := (1 - g[0])/(1 - x); CoefficientList[ Series[b[x], {x, 0, max}], x]*Range[1, max + 1]! (* _Jean-François Alcover_, Jan 24 2013, after _Sergei N. Gladkovskii_ *)
%K easy,nonn
%O 1,2
%A _Vladeta Jovovic_, Sep 22 2007
%E More terms from _R. J. Mathar_, Sep 28 2007
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