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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000332, A005894, A006522.

Programs

GAP

List([0..40], n-> Binomial(n+5,5) - Binomial(n,5)); # G. C. Greubel, Nov 08 2019

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

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 *)

Sage

[binomial(n+5,5) - binomial(n,5) for n in (0..40)] # G. C. Greubel, Nov 08 2019

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