%I #38 Sep 08 2022 08:44:35
%S 1,8,36,120,330,792,1716,3431,6427,11404,19328,31494,49596,75804,
%T 112848,164109,233717,326656,448876,607412,810510,1067760,1390236,
%U 1790643,2283471,2885156,3614248,4491586,5540480,6786900,8259672
%N 6-dimensional centered tetrahedral numbers.
%C 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
%H Vincenzo Librandi, <a href="/A008500/b008500.txt">Table of n, a(n) for n = 0..10000</a>
%H E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 227.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F G.f.: (1-x^7)/(1-x)^8 = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 )/(1-x)^7.
%F a(-1-n) = a(n).
%F 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
%p seq(binomial(n+7,7) - binomial(n,7), n=0..30); # _G. C. Greubel_, Nov 09 2019
%t Table[Binomial[n+7,7] - Binomial[n,7], {n,0,30}] (* _G. C. Greubel_, Nov 09 2019 *)
%o (PARI) a(n)=binomial(n+7,7)-binomial(n,7)
%o (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
%o (Sage) b=binomial; [b(n+7,7) - b(n,7) for n in (0..30)] # _G. C. Greubel_, Nov 09 2019
%o (GAP) B:=Binomial;; List([0..30], n-> B(n+7,7)-B(n,7) ); # _G. C. Greubel_, Nov 09 2019
%Y Partial sums of A008489.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_