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

0,3

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).

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) = 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

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 *)

Prog

(Sage) [(binomial(binomial(n, 3), 2)) for n in range(2, 34)] # Zerinvary Lajos, Nov 30 2009
(Magma) [n*(n^2 - 1)*(n + 2)*(n^2 + 4*n + 6)/72: n in [0..40]]; // Vincenzo Librandi, Sep 21 2011
(PARI) a(n)=n*(n^2-1)*(n+2)*(n^2+4*n+6)/72 \\ Charles R Greathouse IV, Feb 19 2017

Crossrefs

Sequence in context: A123141 A122096 A302709 * A288835 A162230 A258350

Adjacent sequences: A054560 A054561 A054562 * A054564 A054565 A054566

Keyword

easy,nonn,nice

Author

Vladeta Jovovic, Apr 10 2000

Extensions

More terms from James A. Sellers, Apr 11 2000

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

Status

approved