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A028844 Row sums of triangle A013988.

%I A028844 #22 Jan 20 2025 09:02:29
%S A028844 1,6,71,1261,29906,887751,31657851,1318279586,62783681421,
%T A028844 3365947782611,200610405843926,13157941480889921,941848076798467801,
%U A028844 73060842413607398806,6105266987293752470991,546770299628690541571901,52244284936267317229542466,5305131708827069245129523591
%N A028844 Row sums of triangle A013988.
%H A028844 G. C. Greubel, <a href="/A028844/b028844.txt">Table of n, a(n) for n = 1..250</a>
%H A028844 Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%F A028844 E.g.f.: exp(1 - (1-6*x)^(1/6)) - 1.
%F A028844 D-finite with recurrence: a(n) = 15*(2*n-7)*a(n-1) +5*(72*n^2-576*n+1169)*a(n-2) +45*(2*n-9)*(24*n^2-216*n+497)*a(n-3) -20*(324*n^4-6480*n^3+48735*n^2-163350*n+205877)*a(n-4) +12*(6*n-35)*(6*n-31)*(3*n-16)*(2*n-11)*(3*n-17)*a(n-5) +a(n-6). - _R. J. Mathar_, Jan 28 2020
%F A028844 From _Seiichi Manyama_, Jan 20 2025: (Start)
%F A028844 a(n) = Sum_{k=0..n} (-1)^k * 6^(n-k) * |Stirling1(n,k)| * A000587(k).
%F A028844 a(n) = e * (-6)^n * n! * Sum_{k>=0} (-1)^k * binomial(k/6,n)/k!. (End)
%t A028844 With[{nn=20},Rest[CoefficientList[Series[Exp[1-(1-6x)^(1/6)]-1,{x,0,nn}], x]Range[0,nn]!]] (* _Harvey P. Dale_, Feb 02 2012 *)
%o A028844 (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!(Laplace( Exp(1-(1-6*x)^(1/6)) -1 ))); // _G. C. Greubel_, Oct 03 2023
%o A028844 (SageMath)
%o A028844 def A028844_list(prec):
%o A028844     P.<x> = PowerSeriesRing(QQ, prec)
%o A028844     return P( exp(1-(1-6*x)^(1/6)) -1 ).egf_to_ogf().list()
%o A028844 a=A028844_list(40); a[1:] # _G. C. Greubel_, Oct 03 2023
%Y A028844 Sequences with e.g.f. exp(1-(1-m*x)^(1/m)) - 1: A000012 (m=1), A001515 (m=2), A015735 (m=3), A016036 (m=4), A028575 (m=5), this sequence (m=6).
%K A028844 nonn
%O A028844 1,2
%A A028844 _Wolfdieter Lang_