A008500 6-dimensional centered tetrahedral numbers.

1, 8, 36, 120, 330, 792, 1716, 3431, 6427, 11404, 19328, 31494, 49596, 75804, 112848, 164109, 233717, 326656, 448876, 607412, 810510, 1067760, 1390236, 1790643, 2283471, 2885156, 3614248, 4491586, 5540480, 6786900, 8259672

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

If X is an n-set and Y a fixed 7-subset of X then a(n-7) is equal to the number of 7-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 227.
• Milan Janjic, Two Enumerative Functions
• Index entries for two-way infinite sequences
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Partial sums of A008489.

Programs

GAP

B:=Binomial;; List([0..30], n-> B(n+7,7)-B(n,7) ); # G. C. Greubel, Nov 09 2019

Magma

[(720 + 1764*n +735*n^3 +2128*n^2 +385*n^4 +21*n^5 + 7*n^6)/720: n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

Maple

seq(binomial(n+7,7) - binomial(n,7), n=0..30); # G. C. Greubel, Nov 09 2019

Mathematica

Table[Binomial[n+7,7] - Binomial[n,7], {n,0,30}] (* G. C. Greubel, Nov 09 2019 *)

PARI

a(n)=binomial(n+7,7)-binomial(n,7)

Sage

b=binomial; [b(n+7,7) - b(n,7) for n in (0..30)] # G. C. Greubel, Nov 09 2019

Formula

G.f.: (1-x^7)/(1-x)^8 = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 )/(1-x)^7.

a(-1-n) = a(n).

720*a(n) = 720 + 1764*n + 2128*n^2 + 735*n^3 + 385*n^4 + 21*n^5 + 7*n^6. - R. J. Mathar, Mar 14 2011