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
a(n) = (1/2)*Sum_{1 <= i, j <= n+1} (i - j)^4 - Peter Bala, Jun 11 2024
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 *)
Nest[Accumulate[#]&, Range[30]^4, 2] (* Harvey P. Dale, Aug 13 2024 *)
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
KEYWORD
nonn,easy
AUTHOR
Cecilia Rossiter, Dec 14 2004
EXTENSIONS
Edited by Ralf Stephan, Dec 16 2004
STATUS
approved