OFFSET
0,2
COMMENTS
If Y is a 3-subset of an n-set X then, for n >= 8, a(n-8) is the number of 8-subsets of X having at least two elements in common with Y. - Milan Janjic, Nov 23 2007
a(n) is the n-th antidiagonal sum of the convolution array A213551. - Clark Kimberling, Jun 17 2012
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = binomial(n+5, 5)*(n+2)/2.
G.f.: (1+2*x)/(1-x)^7.
From Amiram Eldar, Jan 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 1205/18 - 20*Pi^2/3.
Sum_{n>=0} (-1)^n/a(n) = 10*Pi^2/3 - 320*log(2)/3 + 755/18. (End)
EXAMPLE
From the third formula: a(4) = 15+60+108+120+75 = 378. - Bruno Berselli, Sep 04 2013
MATHEMATICA
CoefficientList[Series[(1 + 2 x)/(1 - x)^7, {x, 0, 25}], x] (* Harvey P. Dale, Mar 13 2011 *)
Nest[Accumulate, Range[1, 120, 3], 5] (* Vladimir Joseph Stephan Orlovsky, Jan 28 2012 *)
Table[Binomial[n + 5, 5] (n + 2) / 2, {n, 0, 35}] (* Vincenzo Librandi, Dec 27 2018 *)
PROG
(Magma) [Binomial(n+5, 5)*(n+2)/2: n in [0..40]]; // Vincenzo Librandi, Dec 27 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Barry E. Williams, Dec 19 1999
STATUS
approved