A008501 7-dimensional centered tetrahedral numbers.

1, 9, 45, 165, 495, 1287, 3003, 6435, 12869, 24301, 43713, 75417, 125475, 202203, 316767, 483879, 722601, 1057265, 1518517, 2144493, 2982135, 4088655, 5533155, 7398411, 9782829, 12802581, 16593929

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

If X is an n-set and Y a fixed 8-subset of X then a(n-8) is equal to the number of 8-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Milan Janjic, Two Enumerative Functions
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Programs

GAP

B:=Binomial;; List([0..30], n-> B(n+8,8)-B(n,8) ); # G. C. Greubel, Nov 09 2019

Magma

[((2*n+1)*(n^6+3*n^5 +100*n^4 +195*n^3 +1159*n^2 +1062*n +1260)/1260) : n in [0..30]]; // Vincenzo Librandi, Oct 08 2011

Maple

seq(binomial(n+8,8) - binomial(n,8), n=0..30); # G. C. Greubel, Nov 09 2019

Mathematica

Table[Binomial[n + 8, 8] - Binomial[n, 8], {n, 0, 26}] (* Bruno Berselli, Mar 22 2012 *)

PARI

vector(31, n, b=binomial; b(n+7,8) - b(n-1,8) ) \\ G. C. Greubel, Nov 09 2019

Sage

b=binomial; [b(n+8,8) - b(n,8) for n in (0..30)] # G. C. Greubel, Nov 09 2019

Formula

G.f.: (1-x^8)/(1-x)^9 = (1+x)*(1+x^2)*(1+x^4)/(1-x)^8.

1260*a(n) = (2*n+1)*(n^6 + 3*n^5 + 100*n^4 + 195*n^3 + 1159*n^2 + 1062*n + 1260). - R. J. Mathar, Mar 14 2011