A001555 a(n) = 1^n + 2^n + ... + 8^n.

8, 36, 204, 1296, 8772, 61776, 446964, 3297456, 24684612, 186884496, 1427557524, 10983260016, 84998999652, 660994932816, 5161010498484, 40433724284976, 317685943157892, 2502137235710736, 19748255868485844, 156142792528260336, 1236466399775623332

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Conjectures for o.g.f.s for this type of sequence appear in the PhD thesis by Simon Plouffe. See A001552 for the reference. These conjectures are proved in a link given in A196837. [Wolfdieter Lang, Oct 15 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 and Robert Israel, Table of n, a(n) for n = 0..1000 (n = 0..200 from T. D. Noe)
• 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 368
• Index entries for linear recurrences with constant coefficients, signature (36, -546, 4536, -22449, 67284, -118124, 109584, -40320).

Crossrefs

Column 8 of array A103438.

Cf. A001018, A001552, A001554, A196837.

Programs

Maple

seq(add(j^n,j=1..8), n=0..20); # Robert Israel, Aug 23 2015

Mathematica

Table[Total[Range[8]^n], {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *)

PARI

first(m)=vector(m,n,n--;sum(i=1,8,i^n)) \\ Anders Hellström, Aug 23 2015

Formula

From Wolfdieter Lang, Oct 15 2011 (Start)

E.g.f.: (1-exp(8*x))/(exp(-x)-1) = Sum_{j=1..8} exp(j*x) (trivial).

O.g.f.: 4*(2-9*x)*(1-27*x+288*x^2-1539*x^3+4299*x^4-5886*x^5+3044*x^6) / Product_{j=1..8} (1-j*x). From the e.g.f. via Laplace transformation. See the proof in a link under A196837. (End)

a(n) = A001554(n) + A001018(n). - Michel Marcus, Jul 26 2013

Extensions

More terms from Jon E. Schoenfield, Mar 24 2010