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A001552
a(n) = 1^n + 2^n + ... + 5^n.
(Formerly M3863 N1584)
11
5, 15, 55, 225, 979, 4425, 20515, 96825, 462979, 2235465, 10874275, 53201625, 261453379, 1289414505, 6376750435, 31605701625, 156925970179, 780248593545, 3883804424995, 19349527020825, 96470431101379, 481245667164585, 2401809362313955, 11991391850823225
OFFSET
0,1
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). - Wolfdieter Lang, Oct 10 2011
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).
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].
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
FORMULA
a(n) = Sum_{k=1..5} k^n, n >= 0.
O.g.f.: (5 - 60*x + 255*x^2 - 450*x^3 + 274*x^4)/Product_{j=1..5} (1 - j*x). - Simon Plouffe in his 1992 dissertation
E.g.f.: exp(x)*(1-exp(5*x))/(1-exp(x)) = Sum_{j=1..5} exp(j*x) (trivial). - Wolfdieter Lang, Oct 10 2011
MATHEMATICA
Table[Total[Range[5]^n], {n, 0, 40}] (* T. D. Noe, Oct 10 2011 *)
PROG
(PARI) a(n)=if(n<0, 0, sum(k=1, 5, k^n))
(Sage) [3**n + sigma(4, n) + 5**n for n in range(22)] # Zerinvary Lajos, Jun 04 2009
(Sage) [1 + 2**n + 3**n + 4**n + 5**n for n in range(22)] # Zerinvary Lajos, Jun 04 2009
CROSSREFS
Column 5 of array A103438.
Sequence in context: A149587 A149588 A137959 * A165731 A309117 A307349
KEYWORD
nonn,easy,changed
STATUS
approved