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Search: id:A000538
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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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+0
47
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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; listen)
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OFFSET
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0,3
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REFERENCES
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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.
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.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 222.
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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, December 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.
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FORMULA
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n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30.
G.f.: x*(1+11*x+11*x^2+x^3)/(1-x)^6. More generally, g.f. for Sum_{k=0..n} k^m is Euler(m, x)/(1-x)^(m+2), where Euler(m, x) is Eulerian polynomial of degree m (cf. A008292). E.g.f.: 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.yu), May 08 2002
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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; [Conjectured by S. Plouffe in his 1992 dissertation.]
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
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A000217, A000330, A000537, A000539, A000540, A000541, A000542, A007487, A023002, A064538.
Cf. A101089.
Row 4 of array A103438.
Sequence in context: A008514 A044268 A044649 this_sequence A023873 A098997 A139497
Adjacent sequences: A000535 A000536 A000537 this_sequence A000539 A000540 A000541
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KEYWORD
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nonn,easy,nice
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AUTHOR
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njas
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