%I #103 Aug 13 2024 12:28:22

%S 1,18,116,470,1449,3724,8400,17172,32505,57838,97812,158522,247793,

%T 375480,553792,797640,1125009,1557354,2120020,2842686,3759833,4911236,

%U 6342480,8105500,10259145,12869766,16011828,19768546,24232545,29506544

%N Second partial sums of fourth powers (A000583).

%C a(n) is the n-th antidiagonal sum of the convolution array A213553. - _Clark Kimberling_, Jun 17 2012

%C a(n-1)/n^5 is the "retention" of water on a 3 X 3 random surface of n levels - see Knecht et al., 2012, Schrenk et al., 2014. - _Robert M. Ziff_, Mar 08 2014

%C The general formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) is the m-th Faulhaber’s polynomial. - _Luciano Ancora_, Jan 26 2015

%H Vincenzo Librandi, <a href="/A101089/b101089.txt">Table of n, a(n) for n = 1..1000</a>

%H Luciano Ancora, <a href="/A253636/a253636_3.pdf">Recurrence relation for the second partial sums of m-th powers</a>.

%H Luciano Ancora, <a href="/A253636/a253636_4.pdf">Second partial sums of the m-th powers</a>.

%H Craig L. Knecht, Walter Trump, Daniel ben-Avraham and Robert M. Ziff, <a href="http://dx.doi.org/10.1103/PhysRevLett.108.045703">Retention Capacity of Random Surfaces</a>, Phys. Rev. Lett., Vol. 108 (2012), 045703.

%H C. P. Neuman and D. I. Schonbach, <a href="http://dx.doi.org/10.1137/1019006">Evaluation of sums of convolved powers using Bernoulli numbers</a>, SIAM Rev., Vol. 19, No. 1 (1977), pp. 90-99. MR0428678 (55 #1698). See Table 1. - _N. J. A. Sloane_, Mar 23 2014

%H Claudio de J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq., Vol. 16 (2013), Article 13.5.7.

%H Cecilia Rossiter, <a href="http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Dead link]

%H Cecilia Rossiter, <a href="/A101089/a101089_1.pdf">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>. [Cached copy, May 15 2013]

%H K. J. Schrenk, N. A. M. Araújo, R. M. Ziff and H. J. Herrmann <a href="http://arxiv.org/abs/1403.2082">Retention Capacity of Correlated Surfaces</a>, arXiv:1403.2082 [cond-mat.stat-mech], 2014.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = (1/60)*n*(n+1)^2*(n+2)*(2*n*(n+2)-1).

%F G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^7. - _Colin Barker_, Apr 04 2012

%F a(n) = Sum_{i=1..n} i*(n+1-i)^4, by the definition. - _Bruno Berselli_, Jan 31 2014

%F a(n) = 2*a(n-1) - a(n-2) + n^4. - _Luciano Ancora_, Jan 08 2015

%F Sum_{n>=1} 1/a(n) = 85/3 + 10*Pi^2/3 - 20*sqrt(2/3)*Pi*cot(sqrt(3/2)*Pi). - _Amiram Eldar_, Jan 26 2022

%F a(n) = (1/2)*Sum_{1 <= i, j <= n+1} (i - j)^4 - _Peter Bala_, Jun 11 2024

%e a(7) = 8400 = 1*(8-1)^4 + 2*(8-2)^4 + 3*(8-3)^4 + 4*(8-4)^4 + 5*(8-5)^4 + 6*(8-6)^4 + 7*(8-7)^4. - _Bruno Berselli_, Jan 31 2014

%p f:=n->(2*n^6-5*n^4+3*n^2)/60;

%p [seq(f(n),n=0..50)]; # _N. J. A. Sloane_, Mar 23 2014

%t a[n_] := n(n+1)^2(n+2)(2n(n+2) -1)/60; Table[a[n], {n, 40}]

%t CoefficientList[Series[(1+x)*(1+10*x+x^2)/(1-x)^7, {x,0,40}], x] (* _Vincenzo Librandi_, Mar 24 2014 *)

%t Nest[Accumulate[#]&,Range[30]^4,2] (* _Harvey P. Dale_, Aug 13 2024 *)

%o (PARI) a(n)=n*(n+1)^2*(n+2)*(2*n*(n+2)-1)/60 \\ _Charles R Greathouse IV_, Mar 18 2014

%o (Magma) [(1/60)*n*(n+1)^2*(n+2)*(2*n*(n+2)-1): n in [1..40]]; // _Vincenzo Librandi_, Mar 24 2014

%o (Sage) [n*(n+1)^2*(n+2)*(2*n*(n+2)-1)/60 for n in range(1,40)] # _Danny Rorabaugh_, Apr 20 2015

%o (GAP) List([1..40], n-> (n+1)^2*(2*(n+1)^4-5*(n+1)^2+3)/60); # _G. C. Greubel_, Jul 31 2019

%Y Partial sums of A000538.

%Y Cf. A000583, A002415, A101090, A201126, A213553.

%K nonn,easy,changed

%O 1,2

%A Cecilia Rossiter, Dec 14 2004

%E Edited by _Ralf Stephan_, Dec 16 2004