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%I M3397 N1375 #56 Apr 13 2022 13:25:15
%S 4,10,30,100,354,1300,4890,18700,72354,282340,1108650,4373500,
%T 17312754,68711380,273234810,1088123500,4338079554,17309140420,
%U 69107159370,276040692700,1102999460754,4408508961460,17623571298330,70462895745100,281757423024354
%N a(n) = 1^n + 2^n + 3^n + 4^n.
%C From _Wolfdieter Lang_, Oct 10 2011: (Start)
%C a(n) = 2*A196836, n >= 0.
%C a(n)*(-1)^n, n >= 0, gives the z-sequence of the Sheffer triangle A049459 ((signed) 4-restricted Stirling1) which is the inverse Sheffer triangle of A143496 with offset [0,0](4-restricted Stirling2). See the W. Lang link under A006232 for general Sheffer a- and z-sequences. The a-sequence of every (signed) r-restricted Stirling1 number Sheffer triangle is A027641/A027642 (Bernoulli numbers).
%C (End)
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001551/b001551.txt">Table of n, a(n) for n = 0..200</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=364">Encyclopedia of Combinatorial Structures 364</a>
%H C. J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq. 16 (2013) #13.5.7.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%F From _Wolfdieter Lang_, Oct 10 2011: (Start)
%F E.g.f.: (1-exp(4*x))/(exp(-x)-1) = Sum_{j=1..4} exp(j*x) (trivial).
%F O.g.f.: 2*(2-5*x)*(1-5*x+5*x^2)/(product(1-j*x,j=1..4) (via Laplace transformation of the o.g.f., and partial fraction decomposition backwards). See the Maple Program for the o.g.f. conjecture by _Simon Plouffe_. This has now been proved.
%F (End)
%p A001551:=-2*(5*z-2)*(5*z**2-5*z+1)/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[Total[Range[4]^n], {n, 0, 40}] (* _T. D. Noe_, Oct 10 2011 *)
%o (Sage) [3**n + sigma(4, n) for n in range(23)] # _Zerinvary Lajos_, Jun 04 2009
%Y Column 4 of array A103438.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_
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