A054563 a(n) = n*(n^2 - 1)*(n + 2)*(n^2 + 4*n + 6)/72.

0, 0, 6, 45, 190, 595, 1540, 3486, 7140, 13530, 24090, 40755, 66066, 103285, 156520, 230860, 332520, 468996, 649230, 883785, 1185030, 1567335, 2047276, 2643850, 3378700, 4276350, 5364450, 6674031, 8239770, 10100265, 12298320, 14881240

Offset: 0

Author

Vladeta Jovovic, Apr 10 2000

Keywords

• easy
• nonn
• nice

Comments

Number of labeled pure 2-complexes on n nodes with 2 2-simplexes.

References

• L. Berzolari, Allgemeine Theorie der Höheren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B. G. Teubner, 1906. p. 353.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000217, A000292, A373733.

Programs

Magma

[n*(n^2 - 1)*(n + 2)*(n^2 + 4*n + 6)/72: n in [0..40]]; // Vincenzo Librandi, Sep 21 2011

Mathematica

Binomial[Binomial[Range[2,40],3],2] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,0,6,45,190,595,1540},40] (* Harvey P. Dale, Sep 20 2011 *)

PARI

a(n)=n*(n^2-1)*(n+2)*(n^2+4*n+6)/72 \\ Charles R Greathouse IV, Feb 19 2017

Sage

[(binomial(binomial(n,3),2)) for n in range(2, 34)] # Zerinvary Lajos, Nov 30 2009

Formula

C(C(n, 3), 2) = 6*C(n, 4) + 15*C(n, 5) + 10*C(n, 6) = n*(n-1)*(n-2)*(n-3)*(n^2+2)/72 = a(n-2).

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7); a(2)=0, a(3)=0, a(4)=6, a(5)=45, a(6)=190, a(7)=595, a(8)=1540. - Harvey P. Dale, Sep 20 2011

G.f.: -((x^2*(x*(x+3)+6))/(x-1)^7). - Harvey P. Dale, Sep 20 2011

a(n) = (binomial(n+2,3)^2 - binomial(n+2,3))/2, n > 0. - Gary Detlefs, Nov 23 2011

a(n) = Sum_{k=1..3} (-1)^(k+1)*binomial(n+2,3+k)*binomial(n+2,3-k). - Gerry Martens, Oct 11 2022

a(n) = A373733(n) - A000292(n). - J.S. Seneschal, Jul 08 2025

E.g.f.: exp(x)*x^2*(216 + 324*x + 138*x^2 + 21*x^3 + x^4)/72. - Stefano Spezia, Jul 09 2025

Extensions

More terms from James Sellers, Apr 11 2000

Offset changed from 2 to 0 by Vincenzo Librandi, Sep 21 2011