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A001552
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1^n + 2^n + ... + 5^n.
(Formerly M3863 N1584)
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9
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5, 15, 55, 225, 979, 4425, 20515, 96825, 462979, 2235465, 10874275, 53201625, 261453379, 1289414505, 6376750435, 31605701625, 156925970179, 780248593545, 3883804424995, 19349527020825, 96470431101379, 481245667164585, 2401809362313955, 11991391850823225
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| a(n)*(-1)^n, n>=0, gives the z-sequence for the Sheffer triangle A049460 ((signed) 5-restricted Stirling1 numbers), which is the inverse triangle of A193685 (5-restricted Stirling2 numbers). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The a-sequence for each (signed) r-restricted Stirling1 Sheffer triangle is A027641/A027642 (Bernoulli numbers). [From Wolfdieter Lang, Oct 10 2011]
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REFERENCES
| 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
| 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 365
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| a(n) = sum(k^n,k=1..5), n>=0.
From Wolfdieter Lang, Oct 10 2011 (Start)
E.g.f.: exp(x)*(1-exp(5*x))/(1-exp(x)) = sum(exp(j*x),j=1..5) (trivial).
O.g.f.: (5-60*x+255*x^2-450*x^3+274*x^4)/product(1-j*x,j=1..5), from the Laplace transform of the e.g.f., leading to the partial fraction decomposition of the given o.g.f. See the Maple program with the conjecture by S. Plouffe. This has now been proved.
(End)
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MAPLE
| A001552:=-(5-60*z+255*z**2-450*z**3+274*z**4)/(z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[Total[Range[5]^n], {n, 0, 40}] (* T. D. Noe, Oct 10 2011 *)
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PROG
| (PARI) a(n)=if(n<0, 0, sum(k=1, 5, k^n))
(Other) 1. sage: [3^n+sigma(4, n)+sigma(5, n)-1 for n in xrange(0, 22)] # 2. sage: [(1^n+2^n+3^n+4^n+ 5^n) for n in xrange(0, 22)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 04 2009]
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CROSSREFS
| Column 5 of array A103438.
Sequence in context: A149587 A149588 A137959 * A165731 A203294 A149589
Adjacent sequences: A001549 A001550 A001551 * A001553 A001554 A001555
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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