A001554 a(n) = 1^n + 2^n + ... + 7^n.

7, 28, 140, 784, 4676, 29008, 184820, 1200304, 7907396, 52666768, 353815700, 2393325424, 16279522916, 111239118928, 762963987380, 5249352196144, 36210966447236, 250337422025488, 1733857359003860, 12027604452404464, 83544895168776356, 580964060390826448

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Conjectures for o.g.f.s for this type of sequences 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, 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 367
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Index entries for linear recurrences with constant coefficients, signature (28, -322, 1960, -6769, 13132, -13068, 5040).

Crossrefs

Column 7 of array A103438. A196837.

Programs

Magma

[1+2^n+3^n+4^n+5^n+6^n+7^n : n in [0..30]]; // Wesley Ivan Hurt, Jul 15 2014

Maple

A001554:=n->add(i^n, i=1..7): seq(A001554(n), n=0..30); # Wesley Ivan Hurt, Jul 15 2014

Mathematica

Table[Total[Range[7]^n], {n, 0, 20}]

Formula

From Wolfdieter Lang, Oct 15 2011: (Start)

E.g.f.: (1-exp(7*x))/(exp(-x)-1) = Sum_{j=1..7} exp(j*x) (trivial).

O.g.f.: (7 - 168*x + 1610*x^2 - 7840*x^3 + 20307*x^4 - 26264*x^5 + 13068*x^6)/Product_{j=1..7} (1 - j*x). From the e.g.f. via Laplace transformation. See the proof in a link under A196837. (End)

Extensions

More terms from Jon E. Schoenfield, Mar 24 2010