login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A004320 a(n) = n*(n+1)*(n+2)^2/6. 10
0, 3, 16, 50, 120, 245, 448, 756, 1200, 1815, 2640, 3718, 5096, 6825, 8960, 11560, 14688, 18411, 22800, 27930, 33880, 40733, 48576, 57500, 67600, 78975, 91728, 105966, 121800, 139345, 158720, 180048, 203456, 229075, 257040, 287490, 320568, 356421, 395200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Consider the set B(n) = {1,2,3,...n}. Let a(0) = 0. Then a(n) = Sum [ b(i)^2 - b(j)^2] for all i, j = 1 to n, b(i) belongs to B(n). E.g., a(3) = (3^2-1^2) + (3^2-2^2) + (2^2-1^2) = 16. - Amarnath Murthy, Jun 01 2001

Partial sums of A016061. - J. M. Bergot, Jun 18 2013

For n>=3, a(n-2) is the number of permutations of n symbols that 3-commute with an n-cycle (see A233440 for definition). - Luis Manuel Rivera Martínez, Feb 24 2014

a(n) is the sum of all pairs with repetitions allowed drawn from the set of triangular numbers from A000217(0) to A000217(n).  This is similar to A027480 but uses triangular numbers instead of the integers.  Example for n=2: 0+1, 0+3, 1+1, 1+3, 3+3 gives sum of 16=a(2). - J. M. Bergot, Mar 23 2016

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

FORMULA

G.f.: x*(3+x)/(1-x)^5. - Paul Barry, Feb 27 2003

a(n) = (n+2)*A000292(n). - Zerinvary Lajos, May 26 2006

a(n) = A047929(n+2)/6. - Zerinvary Lajos, May 09 2007

a(n) = 5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5). - Wesley Ivan Hurt, Oct 28 2014

Sum_{n>=1} 1/a(n) = Pi^2/2 - 9/2. - Jaume Oliver Lafont, Jul 13 2017

MAPLE

[seq ((n+2)*(binomial(n+2, 3)), n=0..45)]; # Zerinvary Lajos, May 26 2006

MATHEMATICA

Table[n (n + 1) (n + 2)^2/6, {n, 0, 40}] (* Wesley Ivan Hurt, Oct 28 2014 *)

PROG

(MAGMA) [n*(n+1)*(n+2)^2/6: n in [0..40] ]; // Vincenzo Librandi, Aug 19 2011

(PARI) a(n)=n*(n+1)*(n+2)^2/6 \\ Charles R Greathouse IV, Jun 18 2013

CROSSREFS

Cf. A016061, A047929, A233440.

Sequence in context: A172482 A212564 A222843 * A089363 A000574 A041233

Adjacent sequences:  A004317 A004318 A004319 * A004321 A004322 A004323

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)