A008499 Number of 5-dimensional centered tetrahedral numbers.

1, 7, 28, 84, 210, 462, 923, 1709, 2975, 4921, 7798, 11914, 17640, 25416, 35757, 49259, 66605, 88571, 116032, 149968, 191470, 241746, 302127, 374073, 459179, 559181, 675962, 811558, 968164, 1148140

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Binomial transform of (1,6,15,20,15,6,0,0,0,...). - Paul Barry, Jul 01 2003

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

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 linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Programs

GAP

B:=Binomial;; List([0..30], n-> B(n+6,6)-B(n,6) ); # G. C. Greubel, Nov 09 2019

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

Maple

seq(binomial(n+6,6) - binomial(n,6), n=0..30); # G. C. Greubel, Nov 09 2019

Mathematica

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 *)

PARI

vector(31, n, b=binomial; b(n+5,6) - b(n-1,6) ) \\ G. C. Greubel, Nov 09 2019

Sage

b=binomial; [b(n+6,6) - b(n,6) for n in (0..30)] # G. C. Greubel, Nov 09 2019

Formula

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

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

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