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

Author

Cecilia Rossiter, Dec 14 2004

Keywords

• nonn
• easy

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

Examples

			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
		

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.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 13.
• 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).

Crossrefs

Partial sums of A000538.

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

Programs

GAP

List([1..40], n-> (n+1)^2*(2*(n+1)^4-5*(n+1)^2+3)/60); # G. C. Greubel, Jul 31 2019

Magma

[(1/60)*n*(n+1)^2*(n+2)*(2*n*(n+2)-1): n in [1..40]]; // Vincenzo Librandi, Mar 24 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 *)
Nest[Accumulate[#]&,Range[30]^4,2] (* Harvey P. Dale, Aug 13 2024 *)

PARI

a(n)=n*(n+1)^2*(n+2)*(2*n*(n+2)-1)/60 \\ Charles R Greathouse IV, Mar 18 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

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

a(n) = (1/2)*Sum_{1 <= i, j <= n+1} (i - j)^4 - Peter Bala, Jun 11 2024

Extensions

Edited by Ralf Stephan, Dec 16 2004