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A033488 a(n) = n*(n+1)*(n+2)*(n+3)/6.

%I A033488 #56 Jul 11 2025 08:47:04
%S A033488 0,4,20,60,140,280,504,840,1320,1980,2860,4004,5460,7280,9520,12240,
%T A033488 15504,19380,23940,29260,35420,42504,50600,59800,70200,81900,95004,
%U A033488 109620,125860,143840,163680,185504,209440,235620,264180,295260,329004,365560,405080
%N A033488 a(n) = n*(n+1)*(n+2)*(n+3)/6.
%C A033488 With two initial 0, convolution of the oblong numbers (A002378) with the nonnegative even numbers (A005843). - _Bruno Berselli_, Oct 24 2016
%H A033488 Vincenzo Librandi, <a href="/A033488/b033488.txt">Table of n, a(n) for n = 0..700</a>
%F A033488 a(n) = n*C(3+n, 3). - _Zerinvary Lajos_, Jan 10 2006
%F A033488 G.f.: 4*x/(1-x)^5. - _Colin Barker_, Mar 01 2012
%F A033488 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 A033488 From _Amiram Eldar_, Jun 02 2022: (Start)
%F A033488 Sum_{n>=1} 1/a(n) = 1/3.
%F A033488 Sum_{n>=1} (-1)^(n+1)/a(n) = 8*log(2) - 16/3. (End)
%F A033488 E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/6. - _Stefano Spezia_, Jul 11 2025
%p A033488 [seq(4*binomial(n+3, 4), n=0..35)]; # _Zerinvary Lajos_, Nov 24 2006
%t A033488 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 A033488 # Binomial[#+3,3]&/@ Range[0,40]  (* _Harvey P. Dale_, Feb 20 2011 *)
%o A033488 (Magma) [n*(n+1)*(n+2)*(n+3)/6: n in [0..40]]; // _Vincenzo Librandi_, Apr 28 2011
%o A033488 (Maxima) A033488(n):=n*(n+1)*(n+2)*(n+3)/6$ makelist(A033488(n),n,0,20); /* _Martin Ettl_, Jan 22 2013 */
%Y A033488 1/beta(n, 4) in A061928.
%Y A033488 Cf. A000332, A034827, A050534.
%Y A033488 Cf. A002378, A005843.
%Y A033488 Convolution of the oblong numbers with the odd numbers: A008911.
%Y A033488 Fourth column of A003506.
%K A033488 nonn,easy
%O A033488 0,2
%A A033488 _N. J. A. Sloane_