A001551 a(n) = 1^n + 2^n + 3^n + 4^n. (Formerly M3397 N1375)

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

0,1

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

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(1-j*x,j=1..4) (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)

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 *)

Prog

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

Crossrefs

Column 4 of array A103438.

Sequence in context: A006357 A114946 A243793 * A363509 A067142 A145453

Adjacent sequences: A001548 A001549 A001550 * A001552 A001553 A001554

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved