A054559 Number of labeled pure 2-complexes on n nodes (0-simplexes) with 5 2-simplexes and 8 1-simplexes.

30, 180, 630, 1680, 3780, 7560, 13860, 23760, 38610, 60060, 90090, 131040, 185640, 257040, 348840, 465120, 610470, 790020, 1009470, 1275120, 1593900, 1973400, 2421900, 2948400, 3562650, 4275180, 5097330, 6041280, 7120080, 8347680, 9738960, 11309760, 13076910

Offset: 5

Author

Vladeta Jovovic, Apr 10 2000

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A054557.

Cf. A000389, A052787.

Programs

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

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 *)

PARI

x='x+O('x^30); Vec(serlaplace(x^5*exp(x)/4)) \\ G. C. Greubel, Nov 23 2017

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)