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A000538
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Sum of fourth powers: 0^4+1^4+...+n^4.
(Formerly M5043 N2179)
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54
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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 S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein, MathWorld: Faulhaber's Formula
Wikipedia, Faulhaber's formula
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
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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
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MAPLE
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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 (zerinvarylajos(AT)yahoo.com), Feb 22 2008
s = 0; lst = {s}; Do[s += n^4; AppendTo[lst, s], {n, 1, 33, 1}]; lst [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 12 2009]
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MATHEMATICA
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Accumulate[Range[0, 40]^4] [From Harvey P. Dale, Jan 13 2011]
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PROG
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(Sage) [bernoulli_polynomial(n, 5)/5 for n in xrange(1, 35)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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
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CROSSREFS
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Cf. A000217, A000330, A000537, A000539, A000540, A000541, A000542, A007487, A023002, A064538, A101089.
Row 4 of array A103438.
Cf. A000583.
Sequence in context: A044268 A044649 A160827 * A023873 A098997 A139497
Adjacent sequences: A000535 A000536 A000537 * A000539 A000540 A000541
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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The general V. Jovovic formula has been slightly changed after his approval by Wolfdieter Lang, Nov 03 2011.
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STATUS
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approved
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