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A001551
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1^n + 2^n + 3^n + 4^n.
(Formerly M3397 N1375)
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7
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4, 10, 30, 100, 354, 1300, 4890, 18700, 72354, 282340, 1108650, 4373500, 17312754, 68711380, 273234810, 1088123500, 4338079554, 17309140420, 69107159370, 276040692700, 1102999460754, 4408508961460, 17623571298330, 70462895745100, 281757423024354
(list;
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OFFSET
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0,1
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COMMENTS
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From Wolfdieter Lang, Oct 10 2011 (Start)
a(n)=2*A196836, n>=0.
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).
(End)
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REFERENCES
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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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 364
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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From Wolfdieter Lang, Oct 10 2011 (Start)
E.g.f.: (1-exp(4*x))/(exp(-x)-1) = sum(exp(j*x),j=1..4) (trivial).
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.
(End)
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MAPLE
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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.]
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MATHEMATICA
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Table[Total[Range[4]^n], {n, 0, 40}] (* T. D. Noe, Oct 10 2011 *)
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PROG
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(Sage) [3^n+sigma(4, n)for n in xrange(0, 23)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]
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CROSSREFS
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Column 4 of array A103438.
Sequence in context: A006357 A047088 A114946 * A067142 A145453 A034730
Adjacent sequences: A001548 A001549 A001550 * A001552 A001553 A001554
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James A. Sellers, Sep 08 2000
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STATUS
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approved
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