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

0, 4, 20, 60, 140, 280, 504, 840, 1320, 1980, 2860, 4004, 5460, 7280, 9520, 12240, 15504, 19380, 23940, 29260, 35420, 42504, 50600, 59800, 70200, 81900, 95004, 109620, 125860, 143840, 163680, 185504, 209440, 235620, 264180, 295260, 329004, 365560, 405080

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

With two initial 0, convolution of the oblong numbers (A002378) with the nonnegative even numbers (A005843). - Bruno Berselli, Oct 24 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..700

Crossrefs

1/beta(n, 4) in A061928.

Cf. A000332, A034827, A050534.

Cf. A002378, A005843.

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

Fourth column of A003506.

Programs

Magma

[n*(n+1)*(n+2)*(n+3)/6: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

Maple

[seq(4*binomial(n+3, 4), n=0..35)]; # Zerinvary Lajos, Nov 24 2006

Mathematica

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 *)
# Binomial[#+3,3]&/@ Range[0,40]  (* Harvey P. Dale, Feb 20 2011 *)

Maxima

A033488(n):=n*(n+1)*(n+2)*(n+3)/6$ makelist(A033488(n),n,0,20); /* Martin Ettl, Jan 22 2013 */

Formula

a(n) = n*C(3+n, 3). - Zerinvary Lajos, Jan 10 2006

G.f.: 4*x/(1-x)^5. - Colin Barker, Mar 01 2012

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

From Amiram Eldar, Jun 02 2022: (Start)

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

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

E.g.f.: exp(x)*x*(24 + 36*x + 12*x^2 + x^3)/6. - Stefano Spezia, Jul 11 2025