A256861 a(n) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(n^2 - n + 6)/720.

1, 8, 42, 168, 546, 1512, 3696, 8184, 16731, 32032, 58058, 100464, 167076, 268464, 418608, 635664, 942837, 1369368, 1951642, 2734424, 3772230, 5130840, 6888960, 9140040, 11994255, 15580656, 20049498, 25574752, 32356808, 40625376, 50642592, 62706336

Offset: 1

Author

Luciano Ancora, Apr 14 2015

Keywords

• nonn
• easy

Comments

This is the case k = n of b(n,k) = n*(n+1)*(n+2)*(n+3)*(n+4)*(k*(n-1)+6)/120, where b(n,k) is the n-th hypersolid number in 6 dimensions generated from an arithmetical progression with the first term 1 and common difference k (see Sardelis et al. paper).

Links

• D. A. Sardelis and T. M. Valahas, On Multidimensional Pythagorean Numbers, arXiv:0805.4070 [math.GM], 2008.
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A000580.

Cf. similar sequences listed in A256859.

Programs

Mathematica

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (6 - n + n^2)/720, {n, 40}]
Table[Times@@(n+Range[0,4])(n^2-n+6)/720,{n,40}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{1,8,42,168,546,1512,3696,8184},40] (* Harvey P. Dale, Sep 25 2019 *)

PARI

vector(40, n, n*(n+1)*(n+2)*(n+3)*(n+4)*(n^2-n+6)/720) \\ Bruno Berselli, Apr 15 2015

Formula

G.f.: x*(1 + 6*x^2)/(1 - x)^8.

a(n) = 6*A000580(n+4) + A000580(n+6). [Bruno Berselli, Apr 15 2015]