A054563 - OEIS

%I #36 Oct 12 2022 05:20:48

%S 0,0,6,45,190,595,1540,3486,7140,13530,24090,40755,66066,103285,

%T 156520,230860,332520,468996,649230,883785,1185030,1567335,2047276,

%U 2643850,3378700,4276350,5364450,6674031,8239770,10100265,12298320,14881240

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

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

%D 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.

%H Vincenzo Librandi, <a href="/A054563/b054563.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F 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.

%F 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

%F G.f.: -((x^2*(x*(x+3)+6))/(x-1)^7). - _Harvey P. Dale_, Sep 20 2011

%F a(n) = (binomial(n+2,3)^2 - binomial(n+2,3))/2, n > 0. - _Gary Detlefs_, Nov 23 2011

%F a(n) = Sum_{k=1..3} (-1)^(k+1)*binomial(n+2,3+k)*binomial(n+2,3-k). - _Gerry Martens_, Oct 11 2022

%t 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 *)

%o (Sage) [(binomial(binomial(n,3),2)) for n in range(2, 34)] # _Zerinvary Lajos_, Nov 30 2009

%o (Magma) [n*(n^2 - 1)*(n + 2)*(n^2 + 4*n + 6)/72: n in [0..40]]; // _Vincenzo Librandi_, Sep 21 2011

%o (PARI) a(n)=n*(n^2-1)*(n+2)*(n^2+4*n+6)/72 \\ _Charles R Greathouse IV_, Feb 19 2017

%K easy,nonn,nice

%O 0,3

%A _Vladeta Jovovic_, Apr 10 2000

%E More terms from _James A. Sellers_, Apr 11 2000

%E Offset changed from 2 to 0 by _Vincenzo Librandi_, Sep 21 2011