A033488 - OEIS

%I #49 Jul 21 2022 15:53:24

%S 0,4,20,60,140,280,504,840,1320,1980,2860,4004,5460,7280,9520,12240,

%T 15504,19380,23940,29260,35420,42504,50600,59800,70200,81900,95004,

%U 109620,125860,143840,163680,185504,209440

%N a(n) = n*(n+1)*(n+2)*(n+3)/6.

%C With two initial 0, convolution of the oblong numbers (A002378) with the nonnegative even numbers (A005843). - _Bruno Berselli_, Oct 24 2016

%H Vincenzo Librandi, <a href="/A033488/b033488.txt">Table of n, a(n) for n = 0..700</a>

%F a(n) = n*C(3+n, 3). - _Zerinvary Lajos_, Jan 10 2006

%F G.f.: 4*x/(1-x)^5. - _Colin Barker_, Mar 01 2012

%F G.f.: 2*x/(1-x)*W(0), where W(k) = 1 + 1/( 1 - x*(k+2)*(k+4)/(x*(k+2)*(k+4) + (k+1)*(k+2)/W(k+1) )) ); (continued fraction). - _Sergei N. Gladkovskii_, Aug 24 2013

%F From _Amiram Eldar_, Jun 02 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 1/3.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(2) - 16/3. (End)

%p [seq(4*binomial(n+3, 4), n=0..35)]; # _Zerinvary Lajos_, Nov 24 2006

%t f[n_]:=n*(n+1)*(n+2)*(n+3)/6; lst={};Do[AppendTo[lst,f[n]],{n,0,5!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2009 *)

%t # Binomial[#+3,3]&/@ Range[0,40]  (* _Harvey P. Dale_, Feb 20 2011 *)

%o (Magma) [n*(n+1)*(n+2)*(n+3)/6: n in [0..40]]; // _Vincenzo Librandi_, Apr 28 2011

%o (Maxima) A033488(n):=n*(n+1)*(n+2)*(n+3)/6$ makelist(A033488(n),n,0,20); /* _Martin Ettl_, Jan 22 2013 */

%Y 1/beta(n, 4) in A061928.

%Y Cf. A000332, A034827, A050534.

%Y Cf. A002378, A005843.

%Y Convolution of the oblong numbers with the odd numbers: A008911.

%Y Fourth column of A003506.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_