Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I M5043 N2179 #154 Oct 28 2024 06:07:29
%S 0,1,17,98,354,979,2275,4676,8772,15333,25333,39974,60710,89271,
%T 127687,178312,243848,327369,432345,562666,722666,917147,1151403,
%U 1431244,1763020,2153645,2610621,3142062,3756718,4463999,5273999,6197520,7246096,8432017,9768353
%N Sum of fourth powers: 0^4 + 1^4 + ... + n^4.
%C This sequence is related to A000537 by the transform a(n) = n*A000537(n) - Sum_{i=0..n-1} A000537(i). - _Bruno Berselli_, Apr 26 2010
%C 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
%C The number of four dimensional hypercubes in a 4D grid with side lengths n. This applies in general to k dimensions. That is, the number of k-dimensional hypercubes in a k-dimensional grid with side lengths n is equal to the sum of 1^k + 2^k + ... + n^k. - _Alejandro Rodriguez_, Oct 20 2020
%D 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.
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 222.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 155.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991, p. 275.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000538/b000538.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H J. L. Bailey, Jr., <a href="http://dx.doi.org/10.1214/aoms/1177732978">A table to facilitate the fitting of certain logistic curves</a>, Annals Math. Stat., 2 (1931), 355-359.
%H J. L. Bailey, <a href="/A002309/a002309.pdf">A table to facilitate the fitting of certain logistic curves</a>, Annals Math. Stat., 2 (1931), 355-359. [Annotated scanned copy]
%H Bruno Berselli, A description of the transform in Comments lines: website <a href="http://www.lanostra-matematica.org/2008/12/sequenze-numeriche-e-procedimenti.html">Matem@ticamente</a> (in Italian).
%H Stefano Capparelli, <a href="https://books.google.com/books?hl=en&lr=&id=y87LDwAAQBAJ&oi=fnd&pg=PP1&ots=vO1h7m80TC&sig=RJbxo4ndnuh96HVN6wTgeA1gQPE#v=onepage&q=oeis&f=false">Notes on Discrete Math</a>, Società Editrice Esculapio SRL (2019) 3-4.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/FaulhabersFormula.html">MathWorld: Faulhaber's Formula</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Faulhaber's_formula">Faulhaber's formula</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F a(n) = n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30.
%F The preceding formula is due to al-Kachi (1394-1437). - _Juri-Stepan Gerasimov_, Jul 12 2009
%F G.f.: x*(x+1)*(1+10*x+x^2)/(1-x)^6. _Simon Plouffe_ in his 1992 dissertation. 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
%F a(n) = Sum_{i = 1..n} J_4(i)*floor(n/i), where J_4 is A059377. - _Enrique Pérez Herrero_, Feb 26 2012
%F 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
%F a(n) = -Sum_{j=1..4} j*Stirling1(n+1,n+1-j)*Stirling2(n+4-j,n). - _Mircea Merca_, Jan 25 2014
%F Sum_{n>=1} (-1)^(n+1)/a(n) = -30*(4 + 3/cos(sqrt(7/3)*Pi/2))*Pi/7. - _Vaclav Kotesovec_, Feb 13 2015
%F a(n) = (n + 1)*(n + 1/2)*n*(n + 1/2 + sqrt(7/12))*(n + 1/2 - sqrt(7/12))/5, see the Graham et al. reference, p. 275. - _Wolfdieter Lang_, Apr 02 2015
%p A000538 := n-> n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30;
%t Accumulate[Range[0,40]^4] (* _Harvey P. Dale_, Jan 13 2011 *)
%t CoefficientList[Series[x (1 + 11 x + 11 x^2 + x^3)/(1 - x)^6, {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 07 2015 *)
%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 1, 17, 98, 354, 979}, 35] (* _Jean-François Alcover_, Feb 09 2016 *)
%t Table[x^5/5+x^4/2+x^3/3-x/30,{x,40}] (* _Harvey P. Dale_, Jun 06 2021 *)
%o (Sage) [bernoulli_polynomial(n,5)/5 for n in range(1, 35)] # _Zerinvary Lajos_, May 17 2009
%o (Haskell)
%o a000538 n = (3 * n * (n + 1) - 1) * (2 * n + 1) * (n + 1) * n `div` 30
%o -- _Reinhard Zumkeller_, Nov 11 2012
%o (Maxima) A000538(n):=n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30$
%o makelist(A000538(n),n,0,30); /* _Martin Ettl_, Nov 12 2012 */
%o (PARI) a(n) = n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30 \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Python)
%o A000538_list, m = [0], [24, -36, 14, -1, 0, 0]
%o for _ in range(10**2):
%o for i in range(5):
%o m[i+1] += m[i]
%o A000538_list.append(m[-1]) # _Chai Wah Wu_, Nov 05 2014
%o (Python)
%o def A000538(n): return n*(n**2*(n*(6*n+15)+10)-1)//30 # _Chai Wah Wu_, Oct 03 2024
%o (Magma) [n*(1+n)*(1+2*n)*(-1+3*n+3*n^2)/30: n in [0..35]]; // _Vincenzo Librandi_, Apr 04 2015
%o (PARI) concat(0, Vec(x*(1+11*x+11*x^2+x^3)/(1-x)^6 + O(x^100))) \\ _Altug Alkan_, Dec 07 2015
%Y Cf. A000217, A000330, A000537, A000539, A000540, A000541, A000542, A007487, A023002, A064538, A101089.
%Y Row 4 of array A103438.
%Y Cf. A000583.
%Y Cf. A254640.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E The general V. Jovovic formula has been slightly changed after his approval by _Wolfdieter Lang_, Nov 03 2011