A001551 a(n) = 1^n + 2^n + 3^n + 4^n.

4, 10, 30, 100, 354, 1300, 4890, 18700, 72354, 282340, 1108650, 4373500, 17312754, 68711380, 273234810, 1088123500, 4338079554, 17309140420, 69107159370, 276040692700, 1102999460754, 4408508961460, 17623571298330, 70462895745100, 281757423024354

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

From Wolfdieter Lang, Oct 10 2011: (Start)

a(n) = 2*A196836, n >= 0.

a(n)*(-1)^n, n >= 0, gives the z-sequence of the Sheffer triangle A049459 ((signed) 4-restricted Stirling1) which is the inverse Sheffer triangle of A143496 with offset [0,0](4-restricted Stirling2). See the W. Lang link under A006232 for general Sheffer a- and z-sequences. The a-sequence of every (signed) r-restricted Stirling1 number Sheffer triangle is A027641/A027642 (Bernoulli numbers).

(End)

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 364
• C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7.
• 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 (10,-35,50,-24).

Crossrefs

Column 4 of array A103438.

Programs

Maple

A001551:=-2*(5*z-2)*(5*z**2-5*z+1)/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

Table[Total[Range[4]^n], {n, 0, 40}] (* T. D. Noe, Oct 10 2011 *)

Sage

[3**n + sigma(4, n) for n in range(23)]  # Zerinvary Lajos, Jun 04 2009

Formula

From Wolfdieter Lang, Oct 10 2011: (Start)

E.g.f.: (1-exp(4*x))/(exp(-x)-1) = Sum_{j=1..4} exp(j*x) (trivial).

O.g.f.: 2*(2-5*x)*(1-5*x+5*x^2)/(Product_{j=1..4} (1-j*x)) (via Laplace transformation of the o.g.f., and partial fraction decomposition backwards). See the Maple Program for the o.g.f. conjecture by Simon Plouffe. This has now been proved. (End)