A001557 a(n) = 1^n + 2^n + ... + 10^n.

10, 55, 385, 3025, 25333, 220825, 1978405, 18080425, 167731333, 1574304985, 14914341925, 142364319625, 1367428536133, 13202860761145, 128037802953445, 1246324856379625, 12170706132009733, 119179318935377305, 1169842891165484965, 11506994510201252425

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 the 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 370.
• Index entries for linear recurrences with constant coefficients, signature (55, -1320, 18150, -157773, 902055, -3416930, 8409500, -12753576, 10628640, -3628800).

Crossrefs

Column 10 of array A103438. Cf. A196837.

Programs

Mathematica

Table[Total[Range[10]^n], {n, 0, 20}] (* T. D. Noe, Aug 09 2012 *)

Python

def A001557(n): return sum(i**n for i in range(1,11)) # Chai Wah Wu, Oct 24 2024

Formula

a(n) = Sum_{j=1..10} j^n, n >= 0.

E.g.f.: exp(x) + exp(2*x) + exp(3*x) + exp(4*x) + exp(5*x) + exp(6*x) + exp(7*x) + exp(8*x) + exp(9*x) + exp(10*x). - Vladeta Jovovic, May 08 2002

From Wolfdieter Lang, Oct 15 2011: (Start)

O.g.f.: (2 - 11*x) *(5 - 220*x + 4070*x^2 - 41140*x^3 + 247049*x^4 - 896368*x^5 + 1903836*x^6 - 2143152*x^7 + 966240*x^8)/Product_{j=1..10} (1 - j*x).

From the e.g.f. via Laplace transformation. See the proof in a link under A196837.

(End)

a(n) = A001556(n) + A011557(n). - Michel Marcus, Jul 26 2013

Extensions

More terms from Jon E. Schoenfield, Mar 24 2010