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

0,2

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).

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

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 *)

Prog

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

Crossrefs

Sequence in context: A247608 A341136 A166322 * A049018 A008489 A023032

Adjacent sequences: A008496 A008497 A008498 * A008500 A008501 A008502

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved