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 A101089 Second partial sums of fourth powers (A000583). 15
 1, 18, 116, 470, 1449, 3724, 8400, 17172, 32505, 57838, 97812, 158522, 247793, 375480, 553792, 797640, 1125009, 1557354, 2120020, 2842686, 3759833, 4911236, 6342480, 8105500, 10259145, 12869766, 16011828, 19768546, 24232545, 29506544 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) is the n-th antidiagonal sum of the convolution array A213553. - Clark Kimberling, Jun 17 2012 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 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 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 Luciano Ancora, Recurrence relation for the second partial sums of m-th powers Luciano Ancora, Second partial sums of the m-th powers Craig L. Knecht, Walter Trump, Daniel ben-Avraham, and Robert M. Ziff, Retention Capacity of Random Surfaces, Phys. Rev. Lett. 108, 045703, 2012. C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev. 19 (1977), no. 1, 90--99. MR0428678 (55 #1698). See Table 1. - N. J. A. Sloane, Mar 23 2014 C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7 C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Dead link] C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube [Cached copy, May 15 2013] K. J. Schrenk, N. A. M. Araújo, R. M. Ziff, H. J. Herrmann Retention Capacity of Correlated Surfaces, arXiv:1403.2082 [cond-mat.stat-mech], 2014. Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1). FORMULA a(n) = (1/60)*n*(n+1)^2*(n+2)*(2*n*(n+2)-1). G.f.: x*(1+x)*(1+10*x+x^2)/(1-x)^7. - Colin Barker, Apr 04 2012 a(n) = Sum_{i=1..n} i*(n+1-i)^4, by the definition. - Bruno Berselli, Jan 31 2014 a(n) = 2*a(n-1) - a(n-2) + n^4. - Luciano Ancora, Jan 08 2015 EXAMPLE 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 MAPLE f:=n->(2*n^6-5*n^4+3*n^2)/60; [seq(f(n), n=0..50)]; # N. J. A. Sloane, Mar 23 2014 MATHEMATICA a[n_] := n(n+1)^2(n+2)(2n(n+2) -1)/60; Table[a[n], {n, 40}] CoefficientList[Series[(1+x)*(1+10*x+x^2)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 24 2014 *) PROG (PARI) a(n)=n*(n+1)^2*(n+2)*(2*n*(n+2)-1)/60 \\ Charles R Greathouse IV, Mar 18 2014 (MAGMA) [(1/60)*n*(n+1)^2*(n+2)*(2*n*(n+2)-1): n in [1..40]]; // Vincenzo Librandi, Mar 24 2014 (Sage) [n*(n+1)^2*(n+2)*(2*n*(n+2)-1)/60 for n in range(1, 40)] # Danny Rorabaugh, Apr 20 2015 (GAP) List([1..40], n-> (n+1)^2*(2*(n+1)^4-5*(n+1)^2+3)/60); # G. C. Greubel, Jul 31 2019 CROSSREFS Partial sums of A000538. Cf. A101090, A201126. Sequence in context: A251937 A061803 A207103 * A022678 A293878 A044350 Adjacent sequences:  A101086 A101087 A101088 * A101090 A101091 A101092 KEYWORD nonn,easy AUTHOR Cecilia Rossiter, Dec 14 2004 EXTENSIONS Edited by Ralf Stephan, Dec 16 2004 STATUS approved

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Last modified August 18 15:00 EDT 2019. Contains 326106 sequences. (Running on oeis4.)