A112415 - OEIS

%I #26 Sep 04 2022 06:26:24

%S 12,60,180,420,840,1512,2520,3960,5940,8580,12012,16380,21840,28560,

%T 36720,46512,58140,71820,87780,106260,127512,151800,179400,210600,

%U 245700,285012,328860,377580,431520,491040,556512,628320,706860,792540,885780,987012,1096680

%N a(n) = C(1+n,1) * C(2+n,1) * C(4+n,2).

%H Vincenzo Librandi, <a href="/A112415/b112415.txt">Table of n, a(n) for n = 0..680</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F From _R. J. Mathar_, Aug 15 2008: (Start)

%F a(n) = (n+1)*(n+2)*(n+3)*(n+4)/2 = A033486(n+1) = 12*A000332(n+4).

%F O.g.f.: 12/(1-x)^5. (End)

%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=12, a(1)=60, a(2)=180, a(3)=420, a(4)=840. - _Harvey P. Dale_, Jul 24 2011

%F From _Amiram Eldar_, Sep 04 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 1/9.

%F Sum_{n>=0} (-1)^n/a(n) = 8*(3*log(2)-2)/9. (End)

%e n=0: C(1+0,1)*C(2+0,1)*C(4+0,2) = C(1,1)*C(2,1)*C(4,2) = 1*2*6 = 12;

%e n=10: C(1+10,1)*C(2+10,1)*C(4+10,2) = C(11,1)*C(12,1)*C(14,2) = 11*12*91 = 12012.

%t Table[(n+1)(n+2)Binomial[4+n,2],{n,0,30}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{12,60,180,420,840},31] (* _Harvey P. Dale_, Jul 24 2011 *)

%o (Magma) [(n+1)*(n+2)*(n+3)*(n+4)/2: n in [0..40]]; // _Vincenzo Librandi_, Apr 28 2011

%Y Cf. A000332, A033486.

%K easy,nonn

%O 0,1

%A _Zerinvary Lajos_, Dec 09 2005