A008499 - OEIS

%I #31 Sep 08 2022 08:44:35

%S 1,7,28,84,210,462,923,1709,2975,4921,7798,11914,17640,25416,35757,

%T 49259,66605,88571,116032,149968,191470,241746,302127,374073,459179,

%U 559181,675962,811558,968164,1148140

%N Number of 5-dimensional centered tetrahedral numbers.

%C Binomial transform of (1,6,15,20,15,6,0,0,0,...). - _Paul Barry_, Jul 01 2003

%C If X is an n-set and Y a fixed 6-subset of X then a(n-6) is equal to the number of 6-subsets of X intersecting Y. - _Milan Janjic_, Jul 30 2007

%H Vincenzo Librandi, <a href="/A008499/b008499.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/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F G.f.: (1-x^6)/(1-x)^7.

%F a(n) = C(n, 0) + 6*C(n, 1) + 15*C(n, 2) + 15*C(n, 3) + 10*C(n, 4) + 6*C(n, 5); a(n) = C(n+6, 6) - C(n, 6); a(n)=(6*n^5 + 15*n^4 + 160*n^3 + 225*n^2 + 314*n + 120)/120. - _Paul Barry_, Jul 01 2003

%F a(0)=1, a(1)=7, a(2)=28, a(3)=84, a(4)=210, a(5)=462; for n>5, a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - _Harvey P. Dale_, Sep 15 2011

%p seq(binomial(n+6,6) - binomial(n,6), n=0..30); # _G. C. Greubel_, Nov 09 2019

%t Table[Binomial[n+6,6]-Binomial[n,6],{n,0,30}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{1,7,28,84,210,462},30] (* _Harvey P. Dale_, Sep 15 2011 *)

%o (Magma) [(6*n^5+15*n^4+160*n^3+225*n^2+314*n+120)/120: n in [0..40] ]; // _Vincenzo Librandi_, Aug 21 2011

%o (PARI) vector(31, n, b=binomial; b(n+5,6) - b(n-1,6) ) \\ _G. C. Greubel_, Nov 09 2019

%o (Sage) b=binomial; [b(n+6,6) - b(n,6) for n in (0..30)] # _G. C. Greubel_, Nov 09 2019

%o (GAP) B:=Binomial;; List([0..30], n-> B(n+6,6)-B(n,6) ); # _G. C. Greubel_, Nov 09 2019

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_