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A060719 a(0) = 1; a(n+1) = a(n) + Sum_{i=0..n} binomial(n,i)*(a(i)+1).

%I A060719 #17 Sep 08 2022 08:45:03
%S A060719 1,3,9,29,103,405,1753,8279,42293,231949,1357139,8427193,55288873,
%T A060719 381798643,2765917089,20960284293,165729739607,1364153612317,
%U A060719 11665484410113,103448316470743,949739632313501,9013431476894645,88304011710168691
%N A060719 a(0) = 1; a(n+1) = a(n) + Sum_{i=0..n} binomial(n,i)*(a(i)+1).
%H A060719 Michael De Vlieger, <a href="/A060719/b060719.txt">Table of n, a(n) for n = 0..575</a>
%H A060719 A. R. Ashrafi, L. Ghanbari Maman, K. Kavousi, F. Koorepazan Moftakhar, <a href="https://arxiv.org/abs/1911.12232">An Algorithm for Constructing All Supercharacter Theories of a Finite Group</a>, arXiv:1911.12232 [math.GR], 2019.
%F A060719 a(n) = 2*Bell(n+1) - 1. - _Vladeta Jovovic_, Feb 11 2003
%F A060719 Equals the binomial transform of A186021. Also, a(n) = A186021(n+1) - 1. - _Gary W. Adamson_ May 20 2013
%e A060719 a(3) = 29 = (30 - 1) = A186021(4) - 1
%p A060719 A060719 := proc(n) option remember; local i; if n=0 then 1 else A060719(n-1)+add(binomial(n-1,i)*(A060719(i)+1),i=0..n-1); fi; end;
%t A060719 Array[2 BellB[# + 1] - 1 &, 23, 0] (* _Michael De Vlieger_, Feb 12 2020 *)
%o A060719 (PARI) vector(26, n, my(m=n-1); 2*sum(k=0, m+1, stirling(m+1, k, 2)) -1 ) \\ _G. C. Greubel_, Feb 12 2020
%o A060719 (Magma) [2*Bell(n+1) -1: n in [0..25]]; // _G. C. Greubel_, Feb 12 2020
%o A060719 (Sage) [2*bell_number(n+1)-1 for n in (0..25)] # _G. C. Greubel_, Feb 12 2020
%Y A060719 Cf. A000110.
%Y A060719 Cf. A186021.
%K A060719 easy,nonn
%O A060719 0,2
%A A060719 _Frank Ellermann_, Apr 23 2001