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A008498 4-dimensional centered tetrahedral numbers. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A292950 A282845 A050190 * A015640 A138780 A108907

Adjacent sequences:  A008495 A008496 A008497 * A008499 A008500 A008501

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 15 08:37 EST 2019. Contains 329144 sequences. (Running on oeis4.)