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A000538 Sum of fourth powers: 0^4+1^4+...+n^4.
(Formerly M5043 N2179)
55
0, 1, 17, 98, 354, 979, 2275, 4676, 8772, 15333, 25333, 39974, 60710, 89271, 127687, 178312, 243848, 327369, 432345, 562666, 722666, 917147, 1151403, 1431244, 1763020, 2153645, 2610621, 3142062, 3756718, 4463999, 5273999, 6197520, 7246096, 8432017, 9768353 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This sequence is related to A000537 by the transform a(n) = n*A000537(n)-sum(A000537(i), i=0..n-1). - Bruno Berselli, Apr 26 2010

A formula for the r-th successive summation of k^4, for k = 1 to n, is ((12*n^2+(12*n-5)*r+r^2)*(2*n+r)*(n+r)!)/((r+4)!*(n-1)!),(H. W. Gould). - Gary Detlefs, Jan 02 2014

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.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 222.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.

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..1000

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].

B. Berselli, A description of the transform in Comments lines: website Matem@ticamente (in Italian).

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Eric Weisstein, MathWorld: Faulhaber's Formula

Wikipedia, Faulhaber's formula

Index entries for sequences related to linear recurrences with constant coefficients

FORMULA

a(n) = n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30.

The preceding formula is due to al-Kachi (1394-1437). - Juri-Stepan Gerasimov, Jul 12 2009

G.f.: x*(1+11*x+11*x^2+x^3)/(1-x)^6. More generally, the o.g.f. for Sum_{k=0..n} k^m is x*E(m, x)/(1-x)^(m+2), where E(m, x) is the Eulerian polynomial of degree m (cf. A008292). The e.g.f. for these o.g.f.s is: x/(1-x)^2*(exp(y/(1-x))-exp(x*y/(1-x)))/(exp(x*y/(1-x))-x*exp(y/(1-x))). - Vladeta Jovovic, May 08 2002.

a (n) = sum (i = 1 .. n, J_ 4 (i)*floor (n/i)), where J_ 4 is A059377. - Enrique Pérez Herrero, Feb 26 2012

a(n) = 5*a(n-1) - 10* a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + 24. - Ant King, Sep 23 2013

a(n)=-sum(j=1..4, j*s(n+1,n+1-j)*S(n+4-j,n)), where s(n,k) and S(n,k) are the Stirling numbers of the first kind and the second kind, respectively. - Mircea Merca, Jan 25 2014

MAPLE

A000538 := n-> n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30;

A000538:=(1+z)*(z**2+10*z+1)/(z-1)**6; [Simon Plouffe in his 1992 dissertation. Gives sequence without initial zero.]

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=a[n-1]+n^4 od: seq(a[n], n=0..33); - Zerinvary Lajos, Feb 22 2008

s = 0; lst = {s}; Do[s += n^4; AppendTo[lst, s], {n, 1, 33, 1}]; lst [Zerinvary Lajos, Jul 12 2009]

MATHEMATICA

Accumulate[Range[0, 40]^4]  [From Harvey P. Dale, Jan 13 2011]

PROG

(Sage) [bernoulli_polynomial(n, 5)/5 for n in xrange(1, 35)]# [Zerinvary Lajos, May 17 2009]

(Haskell)

a000538 n = (3 * n * (n + 1) - 1) * (2 * n + 1) * (n + 1) * n `div` 30

-- Reinhard Zumkeller, Nov 11 2012

(Maxima) A000538(n):=n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30$

makelist(A000538(n), n, 0, 30); /* Martin Ettl, Nov 12 2012 */

(PARI) a(n) = n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30 \\ Charles R Greathouse IV, Nov 20 2012

CROSSREFS

Cf. A000217, A000330, A000537, A000539, A000540, A000541, A000542, A007487, A023002, A064538, A101089.

Row 4 of array A103438.

Cf. A000583.

Sequence in context: A231683 A231687 A231689 * A023873 A098997 A139497

Adjacent sequences:  A000535 A000536 A000537 * A000539 A000540 A000541

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

The general V. Jovovic formula has been slightly changed after his approval by Wolfdieter Lang, Nov 03 2011.

STATUS

approved

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Last modified April 25 04:22 EDT 2014. Contains 240994 sequences.