

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
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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) = (3^21^2) + (3^22^2) + (2^21^2) = 16.  Amarnath Murthy, Jun 01 2001
Partial sums of A016061.  J. M. Bergot, Jun 18 2013
For n>=3, a(n2) is the number of permutations of n symbols that 3commute with an ncycle (see A233440 for definition).  Luis Manuel Rivera MartÃnez, 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 kcommuting 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)/(1x)^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(n1)10*a(n2)+10*a(n3)5*a(n4)+a(n5).  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



