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%I M3968 N1639 #102 Jan 01 2024 18:19:14
%S 0,1,5,32,288,3413,50069,873612,17650828,405071317,10405071317,
%T 295716741928,9211817190184,312086923782437,11424093749340453,
%U 449317984130199828,18896062057839751444,846136323944176515621
%N a(n) = Sum_{k=1..n} k^k.
%C Starting from the second term, 1, the terms could be described as the special case (n=1; j=1) of the following general formula: a(n) = <from k=j to k=i> Sum [(n + k - 1)]^(k) n=1; j=1; i=1,2,3,...,... For (n=0; j=1) the formula yields A062815 n=0; j=1; i=2,3,4,... For (n=2; j=0) we get A060946 and for (n=3; j=0) A117887. - _Alexander R. Povolotsky_, Sep 01 2007
%C From _Luan Alberto Ferreira_, Aug 01 2017: (Start)
%C If n == 0 or 3 (mod 4), then a(n) == 0 (mod 4).
%C If n == 0, 4, 7, 14, 15 or 17 (mod 18), then a(n) == 0 (mod 3). (End)
%C Called the hypertriangular function by M. K. Azarian. - _Light Ediand_, Nov 19 2021
%D József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, p. 308.
%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="/A001923/b001923.txt">Table of n, a(n) for n=0..100</a>
%H Mohammad K. Azarian, <a href="http://ijpam.eu/contents/2007-36-2/9/9.pdf">On the hyperfactorial function, hypertriangular function and the discriminants of certain polynomials</a>, Int. J. Pure Appl. Math., Vol. 36, No. 2 (2007), pp. 251-257.
%H Andrew Cusumano, <a href="https://www.fq.math.ca/Problems/advanced45-2.pdf">Problem H-656</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 45, No. 2 (2007), p. 187; <a href="https://www.fq.math.ca/Problems/Aug2009advanced.pdf">A Sequence Tending To e</a>, Solution to Problem H-656, ibid., Vol. 46-47, No. 3 (2008/2009), pp. 285-287.
%H G. W. Wishard (proposer) and F. Underwood (solution), <a href="http://www.jstor.org/stable/2306261">Problem 4155: Bound for a Finite Sum</a>, Amer. Math. Monthly, Vol. 53, No. 8 (1946), pp. 471-473.
%F a(n) = A062970(n) - 1.
%F a(n+1)/a(n) > e*n and a(n+1)/a(n) is asymptotic to e*n. - _Benoit Cloitre_, Sep 29 2002
%F For n > 0: a(n) = a(n-1) + A000312(n). - _Reinhard Zumkeller_, Jul 11 2014
%F Limit_{n->oo} (a(n+2)/a(n+1) - a(n+1)/a(n)) = e (Cusumano, 2007). - _Amiram Eldar_, Jan 03 2022
%t Accumulate[Join[{0},Table[k^k,{k,20}]]] (* _Harvey P. Dale_, Feb 11 2015 *)
%o (PARI) for(n=1,20,print1(sum(x=1,n,x^x), ", ")) \\ _Jorge Coveiro_, Dec 24 2004
%o (Haskell)
%o a001923 n = a001923_list !! n
%o a001923_list = scanl (+) 0 $ tail a000312_list
%o -- _Reinhard Zumkeller_, Jul 11 2014
%o (Python) # generates initial segment of sequence
%o from itertools import accumulate
%o def f(k): return 0 if k == 0 else k**k
%o def aupton(nn): return list(accumulate(f(k) for k in range(nn+1)))
%o print(aupton(17)) # _Michael S. Branicky_, Feb 12 2022
%Y Cf. A073825, A062970 (another version).
%Y Cf. A062815, A060946, A117887.
%Y Cf. A000312, A002109.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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