|
| |
|
|
A062392
|
|
a(n)=n^4-(n-1)^4+(n-2)^4-....0^4.
|
|
10
|
|
|
|
0, 1, 15, 66, 190, 435, 861, 1540, 2556, 4005, 5995, 8646, 12090, 16471, 21945, 28680, 36856, 46665, 58311, 72010, 87990, 106491, 127765, 152076, 179700, 210925, 246051, 285390, 329266, 378015, 431985, 491536, 557040, 628881, 707455
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
Number of edges in the join of two complete graphs of order n^2 and n, K_n^2 * K_n - Roberto E. Martinez II (remartin(AT)fas.harvard.edu), Jan 07 2002
a(n) is equal to the partial sums of A007588, stella octangula numbers: n(2n^2 - 1). - Jonathan Vos Post, Mar 15 2006
Contribution from Peter Luschny, Jul 12 2009: (Start)
The general formula for alternating sums of powers is in terms of the Swiss-Knife polynomials P(n,x) A153641 2^(-n-1)(P(n,1)-(-1)^k P(n,2k+1)). Thus
a(k) = |2^(-5)(P(4,1)-(-1)^k P(4,2k+1))|. (End)
|
|
|
REFERENCES
|
T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.
|
|
|
LINKS
|
Harry J. Smith, Table of n, a(n) for n=0,...,1000
Milan Janjic, Two Enumerative Functions
|
|
|
FORMULA
|
a(n) = n(n+1)(n^2+n-1)/2 = n^4-a(n-1) = A000583(n)-a(n) = A000217(A028387(n-1)) = A000217(n)*A028387(n-1).
a(n) = SUM[i=0..n] A007588(i). a(n) = SUM[i=0..n] n*(2*n^2 - 1). a(n) = SUM[i=0..n] (1/6)*(12*n^3-6*n), n>0. - Jonathan Vos Post, Mar 15 2006
a(0)=0, a(1)=1, a(2)=15, a(3)=66, a(4)=190, a(n)=5*a(n-1)-10*a(n-2)+ 10*a(n-3)-5*a(n-4)+a(n-5) [From Harvey P. Dale, Oct 19 2011]
G.f.: -((x*(x*(x+10)+1))/(x-1)^5) [From Harvey P. Dale, Oct 19 2011]
|
|
|
MAPLE
|
a := n -> (2*n^2+n^3-1)*n/2; [From Peter Luschny, Jul 12 2009]
|
|
|
MATHEMATICA
|
k=0; lst={k}; Do[k=n^4-k; AppendTo[lst, k], {n, 1, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Dec 11 2008]
Table[n(n+1)(n^2+n-1)/2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 15, 66, 190}, 40] (* From Harvey P. Dale, Oct 19 2011 *)
|
|
|
PROG
|
(PARI) { a=0; for (n=0, 1000, write("b062392.txt", n, " ", a=n^4 - a) ) } [From Harry J. Smith, Aug 07 2009]
|
|
|
CROSSREFS
|
Cf. A000538, A000583. A062393 provides the result for 5th powers, A011934 for cubes, A000217 for squares, A001057 (unsigned) for nonnegative integers, A000035 (offset) for 0th powers.
Cf. A007588.
Sequence in context: A027526 A033653 A088058 * A211787 A211917 A015876
Adjacent sequences: A062389 A062390 A062391 * A062393 A062394 A062395
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Henry Bottomley, Jun 21 2001
|
|
|
STATUS
|
approved
|
| |
|
|
