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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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51
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) = n*A000537(n) - sum [i = 0 ... n-1] A000537(i) [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Apr 26 2010]
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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).
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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].
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Index entries for sequences related to linear recurrences with constant coefficients
B. Berselli, a description of the recursive method n*Ar(n)-sum[i=0...n-1]Ar(i) (Ar(m) is the m-th term of a sequence): website Matem@ticamente. [From Bruno Berselli (berselli.bruno(AT)yahoo.it), Apr 26 2010]
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FORMULA
| 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 (vladeta(AT)eunet.rs), May 08 2002
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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; [S. 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
| lst={}; s=0; Do[s=s+n^4; AppendTo[lst, s], {n, 10^2}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 14 2008]
Accumulate[Range[0, 40]^4] [From Harvey P. Dale, Jan. 13, 2011]
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PROG
| (Other) sage: [bernoulli_polynomial(n, 5)/5 for n in xrange(1, 35)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
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CROSSREFS
| Cf. A000217, A000330, A000537, A000539, A000540, A000541, A000542, A007487, A023002, A064538.
Cf. A101089.
Row 4 of array A103438.
Sequence in context: A044268 A044649 A160827 * A023873 A098997 A139497
Adjacent sequences: A000535 A000536 A000537 * A000539 A000540 A000541
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| The general V. Jovovic formula has been slightly changed after his approval by Wolfdieter Lang, Nov 03 2011.
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