A001552 a(n) = 1^n + 2^n + ... + 5^n. (Formerly M3863 N1584)

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

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 365

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

Index entries for linear recurrences with constant coefficients, signature (15, -85, 225, -274, 120).

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

Adjacent sequences: A001549 A001550 A001551 * A001553 A001554 A001555

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved