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A008499 Number of 5-dimensional centered tetrahedral numbers. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A073363 A247608 A166322 * A049018 A008489 A023032

Adjacent sequences:  A008496 A008497 A008498 * A008500 A008501 A008502

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 21 00:11 EST 2019. Contains 329348 sequences. (Running on oeis4.)