A004282 a(n) = n*(n+1)^2*(n+2)^2/12.

0, 3, 24, 100, 300, 735, 1568, 3024, 5400, 9075, 14520, 22308, 33124, 47775, 67200, 92480, 124848, 165699, 216600, 279300, 355740, 448063, 558624, 690000, 845000, 1026675, 1238328, 1483524, 1766100, 2090175

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = C(2+n, 2)*C(2+n, 3) = A000217(n+1)*A000292(n). - Zerinvary Lajos, Jan 10 2006

a(n-1) = Sum_{1 <= x_1, x_2 <= n} x_1*(det V(x_1,x_2))^2 = Sum_{1 <= i,j <= n} i*(i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007

G.f.: x*(3+6*x+x^2)/(1-x)^6. - Colin Barker, Feb 09 2012

a(n) = Sum_{k=0..n} Sum_{i=0..n} (n-i+1) * C(k+1,k-1). - Wesley Ivan Hurt, Sep 21 2017

a(n) = A004302(n+1) - A000537(n+1). - J. M. Bergot, Mar 28 2018

From Amiram Eldar, May 29 2022: (Start)

Sum_{n>=1} 1/a(n) = 30 - 3*Pi^2.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 24*log(2) + 12. (End)

Maple

a:= n-> binomial(2+n, 2)*binomial(2+n, 3): seq(a(n), n=0..31); # Zerinvary Lajos, Apr 26 2007

Mathematica

Table[n*(n+1)^2*(n+2)^2/12, {n, 0, 40} (* Vincenzo Librandi, Feb 09 2012 *)

Prog

(Magma) [n*(n+1)^2*(n+2)^2/12: n in [0..50]]; // Vincenzo Librandi, Feb 09 2012
(PARI) a(n) = binomial(n+2, 2)*binomial(n+2, 3); \\ Charles R Greathouse IV, Feb 09 2012

Crossrefs

Cf. A006542, A107891, A114242.

Sequence in context: A274954 A156832 A092780 * A374488 A216725 A334038

Adjacent sequences: A004279 A004280 A004281 * A004283 A004284 A004285

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved