A101089 Second partial sums of fourth powers (A000583).

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

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., Vol. 108 (2012), 045703.

C. P. Neuman and D. I. Schonbach, Evaluation of sums of convolved powers using Bernoulli numbers, SIAM Rev., Vol. 19, No. 1 (1977), pp. 90-99. MR0428678 (55 #1698). See Table 1. - N. J. A. Sloane, Mar 23 2014

Claudio de J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq., Vol. 16 (2013), Article 13.5.7.

Cecilia Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [Dead link]

Cecilia 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 and 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

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

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. A000583, A101090, A201126, A213553.

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