OFFSET
5,1
COMMENTS
Number of {T_1,T_2,...,T_k} where T_i,i=1..k are 3-subsets of an n-set such that {D | D is 2-subset of T_i for some i=1..k} has l elements; k=5,l=8.
Let H be the n X n Hilbert matrix H(i,j) = 1/(i+j-1) for 1 <= i,j <= n. Let B be the inverse matrix of H. The sum of the elements in row 3 of B equals -a(n+2). - T. D. Noe, May 01 2011
REFERENCES
V. Jovovic, On the number of two-dimensional simplicial complexes (in Russian), Metody i sistemy tekhnicheskoy diagnostiki, Vypusk 16, Mezhvuzovskiy zbornik nauchnykh trudov, Izdatelstvo Saratovskogo universiteta, 1991.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 5..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
a(n) = 30*C(n,5) = 30*A000389(n) = n*(n-1)*(n-2)*(n-3)*(n-4)/4.
G.f.: 30*x^5/(1-x)^6. - Colin Barker, Jan 19 2012
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). - Vincenzo Librandi, Apr 29 2012
E.g.f.: x^5*exp(x)/4. - G. C. Greubel, Nov 23 2017
From Amiram Eldar, Mar 08 2022: (Start)
Sum_{n>=5} 1/a(n) = 1/24.
Sum_{n>=5} (-1)^(n+1)/a(n) = 8*log(2)/3 - 131/72. (End)
MATHEMATICA
Table[n*(n+1)*(n+2)*(n+3)*(n+4)/4, {n, 1, 100}] (* Vladimir Joseph Stephan Orlovsky, Jul 21 2009 *)
CoefficientList[Series[30/(1-x)^6, {x, 0, 30}], x] (* Vincenzo Librandi, Apr 29 2012 *)
PROG
(Magma) I:=[30, 180, 630, 1680, 3780, 7560]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+6*Self(n-5)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Apr 29 2012
(PARI) x='x+O('x^30); Vec(serlaplace(x^5*exp(x)/4)) \\ G. C. Greubel, Nov 23 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vladeta Jovovic, Apr 10 2000
STATUS
approved