A193575 T(n)^3 - n^3 where T(n) is a triangular number.

0, 19, 189, 936, 3250, 9045, 21609, 46144, 90396, 165375, 286165, 472824, 751374, 1154881, 1724625, 2511360, 3576664, 4994379, 6852141, 9253000, 12317130, 16183629, 21012409, 26986176, 34312500, 43225975, 53990469, 66901464, 82288486, 100517625, 121994145

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = (n^3*(n^3+3*n^2+3*n-7)/8) = (1/8)*(n-1)*(n^2+4*n+7)*n^3.

From Wesley Ivan Hurt, Aug 23 2014: (Start)

G.f.: x^2*(19+56*x+12*x^2+2*x^3+x^4)/(1-x)^7.

a(n) = 7*a(n-1)-21*a(n-2)+35*a(n-3)-35*a(n-4)+21*a(n-5)-7*a(n-6)+a(n-7).

a(n) = sum_{i=1..n} sum_{j=1..n} sum_{k=1..n} (i*j*k-1).

a(n) = A000217(n)^3 - A000578(n), n > 0.

(End)

Maple

A193575:=n->n^3*(n^3+3*n^2+3*n-7)/8: seq(A193575(n), n=1..40); # Wesley Ivan Hurt, Aug 23 2014

Mathematica

Table[n^3*(n^3 + 3*n^2 + 3*n - 7)/8, {n, 40}] (* Wesley Ivan Hurt, Aug 23 2014 *)
CoefficientList[Series[x*(19 + 56 x + 12 x^2 + 2 x^3 + x^4)/(1 - x)^7, {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 23 2014 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 19, 189, 936, 3250, 9045, 21609}, 40] (* Harvey P. Dale, Oct 24 2020 *)

Prog

(Magma) [(n^3*(n^3+3*n^2+3*n-7)/8): n in [1..40]]

Crossrefs

Cf. A000217, A000578.

Sequence in context: A211866 A327848 A034273 * A161512 A162347 A161879

Adjacent sequences: A193572 A193573 A193574 * A193576 A193577 A193578

Keyword

nonn,easy

Author

Vincenzo Librandi, Sep 08 2011

Status

approved