A008498 4-dimensional centered tetrahedral numbers.

1, 6, 21, 56, 126, 251, 456, 771, 1231, 1876, 2751, 3906, 5396, 7281, 9626, 12501, 15981, 20146, 25081, 30876, 37626, 45431, 54396, 64631, 76251, 89376, 104131, 120646, 139056, 159501, 182126

Offset

0,2

Comments

Binomial transform of (1,5,10,10,5,0,0,0,...). - Paul Barry, Jul 01 2003

If X is an n-set and Y a fixed 5-subset of X then a(n-5) is equal to the number of 5-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007

Also the sum of five consecutive terms of A000332. - Bruno Berselli, Jun 18 2015

References

E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 224 (general formula for n-th centered polytope number).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Milan Janjic, Two Enumerative Functions

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

G.f.: (1-x^5)/(1-x)^6 = (1 +x +x^2 +x^3 +x^4)/(1-x)^5.

a(n) = C(n,0) + 5*C(n,1) + 10*C(n,2) + 10*C(n,3) + 5*C(n,4). - Paul Barry, Jul 01 2003

a(n) = (5*n^4 + 10*n^3 + 55*n^2 + 50*n + 24)/24. - Paul Barry, Jul 01 2003

a(n) = binomial(n+5,5) - binomial(n,5). - Zerinvary Lajos, Jul 21 2006

a(n) = 1 + 5*A006522(n+2). - Bruno Berselli, Jun 18 2015

E.g.f.: (24 + 120*x + 120*x^2 + 40*x^3 + 5*x^4)*exp(x)/24. - G. C. Greubel, Nov 08 2019

Maple

[seq(binomial(n+5, 5)-binomial(n, 5), n=0..45)]; # Zerinvary Lajos, Jul 21 2006

Mathematica

LinearRecurrence[{5, -10, 10, -5, 1}, {1, 6, 21, 56, 126}, 40] (* Harvey P. Dale, Dec 18 2013 *)
Table[1 + 5n(n+1)(n^2 +n +10)/24, {n, 0, 40}] (* Bruno Berselli, Jun 18 2015 *)

Prog

(Magma) [(5*n^4+10*n^3+55*n^2+50*n+24)/24: n in [0..30] ]; // Vincenzo Librandi, Aug 21 2011
(Magma) [Binomial(n+5, 5) - Binomial(n, 5): n in [0..40]]; // G. C. Greubel, Nov 08 2019
(Sage) [binomial(n+5, 5) - binomial(n, 5) for n in (0..40)] # G. C. Greubel, Nov 08 2019
(GAP) List([0..40], n-> Binomial(n+5, 5) - Binomial(n, 5)); # G. C. Greubel, Nov 08 2019

Crossrefs

Cf. A000332, A005894, A006522.

Sequence in context: A282845 A341135 A050190 * A015640 A138780 A108907

Adjacent sequences: A008495 A008496 A008497 * A008499 A008500 A008501

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved