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%I M5043 N2179 #145 Sep 03 2023 12:08:17
%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 B. 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*(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
%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*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
%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;
%p A000538:=(1+z)*(z**2+10*z+1)/(z-1)**6; # _Simon Plouffe_ in his 1992 dissertation; gives sequence without initial zero
%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 (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
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