A001553 a(n) = 1^n + 2^n + ... + 6^n.

6, 21, 91, 441, 2275, 12201, 67171, 376761, 2142595, 12313161, 71340451, 415998681, 2438235715, 14350108521, 84740914531, 501790686201, 2978035877635, 17706908038281, 105443761093411, 628709267031321, 3752628871164355, 22418196307542441, 134023513204581091

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For the o.g.f.s of such sequences see the W. Lang link under A196837. The e.g.f.s are trivial. - Wolfdieter Lang, Oct 14 2011

a(n) is divisible by 7 iff n is not divisible by 6 (see De Koninck & Mercier reference). Example: a(5)= 12201 = 7 * 1743 and a(6) = 67171 = 9595 * 7 + 6. - Bernard Schott, Mar 06 2020

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.
• J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 289 pp. 45, 194, Ellipses, Paris, (2004).
• 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 366
• Index entries for linear recurrences with constant coefficients, signature (21, -175, 735, -1624, 1764, -720).

Crossrefs

Column 6 of array A103438, A001552.

Programs

Mathematica

Table[Total[Range[6]^n], {n, 0, 40}] (* T. D. Noe, Oct 10 2011 *)

Formula

a(n) = Sum_{k=1..6} k^n.

From Wolfdieter Lang, Oct 10 2011: (Start)

E.g.f.: (1-exp(6*x))/(exp(-x)-1) = Sum_{j=1..6} exp(j*x) (trivial).

O.g.f.: (2 - 7*x)*(3 - 42*x + 203*x^2 - 392*x^3 + 252*x^4)/Product_{j=1..6} (1 - j*x).

From the Laplace transformation of the e.g.f. (with argument 1/p, and multiplied with 1/p), which yields the partial fraction decomposition of the given o.g.f., namely Sum_{j=1..6} 1/(1 - j*x).

(End)

Extensions

More terms from Jon E. Schoenfield, Mar 24 2010