A005586 a(n) = n*(n+4)*(n+5)/6.

0, 5, 14, 28, 48, 75, 110, 154, 208, 273, 350, 440, 544, 663, 798, 950, 1120, 1309, 1518, 1748, 2000, 2275, 2574, 2898, 3248, 3625, 4030, 4464, 4928, 5423, 5950, 6510, 7104, 7733, 8398, 9100, 9840, 10619, 11438, 12298, 13200, 14145, 15134, 16168, 17248

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of walks on square lattice.

Number of standard tableaux of shape (n+2,3) (n >= 1). - Emeric Deutsch, May 20 2004

Number of left factors of Dyck paths from (0,0) to (n+5,n-1). E.g. a(1)=5 because we have UDUDUD, UDUUDD, UUDDUD, UUDUDD and UUUDDD, where U=(1,1) and D=(1,-1). - Emeric Deutsch, Jan 25 2005

Column 4 of Catalan triangle A009766. - Zerinvary Lajos, Nov 25 2006

Sum of first n triangular numbers minus next triangular number. - Vladimir Joseph Stephan Orlovsky, Oct 13 2009

Number of packed increasing tableaux of shape 3 X (n+1) with alphabet [n+4]. - Oliver Pechenik, Jan 03 2022

Examples

			G.f. = 5*x + 14*x^2 + 28*x^3 + 48*x^4 + 75*x^5 + 110*x^6 + 154*x^7 + ...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• C. Gaetz, O. Pechenik, J. Striker, and J. P. Swanson, Curious cyclic sieving on increasing tableaux, arXiv:2112.09228 [math.CO], 2021. See Proposition 1.1 at the top of page 2.
• Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988
• Richard K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), Article 00.1.6.
• Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000217, A000292, A005581, A009766, A053121.

a(n)=A053121(n+5,n-1).

Programs

Magma

[n*(n+4)*(n+5)/6: n in [0..50]]; // Vincenzo Librandi, Jun 09 2013

Maple

[seq(binomial(n,3 )-binomial(n,1),n=4..48)]; # Zerinvary Lajos, Nov 25 2006
a:=n->sum ((j-3)*j/2,j=0..n): seq(a(n),n=4..48); # Zerinvary Lajos, Dec 17 2006
A005586:=z*(5-6*z+2*z**2)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation
seq(sum(binomial(n,m), m=1..3)-n^2,n=5..49); # Zerinvary Lajos, Jun 19 2008

Mathematica

Clear[lst,n,a,f]; f[n_]:=n*(n+1)/2; a=0;lst={};Do[a+=f[n];AppendTo[lst,a-f[n+1]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009 *)
CoefficientList[Series[x (5 - 6 x + 2 x^2) / (1 - x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
Table[(n(n+4)(n+5))/6,{n,0,50}] (* or *) LinearRecurrence[{4,-6,4,-1},{0,5,14,28},50] (* Harvey P. Dale, Jul 14 2018 *)

PARI

{a(n) = n * (n+4) * (n+5) / 6}; /* Michael Somos, Apr 13 2007 */

Formula

G.f.: x * (5 - 6*x + 2*x^2) / (1 - x)^4.

E.g.f.: (5*x + 2*x^2 + x^3/6) * exp(x). - Michael Somos, Apr 13 2007

Let t(n) = n*(n+1)/2, te(n) = (n+1)*(n+2)*(n+3)/6. Then a(n-4) = -2*t(n) + te(n-1), e.g., a(2) = -2*t(6) + te(5) = -2*21 + 56 = 14, where te(n) are the tetrahedral numbers A000292 and t(n) are the triangular numbers A000217. - Jon Perry, Jul 23 2003

a(n) = C(5+n, 3)-C(5+n, 2). - Zerinvary Lajos, Jan 09 2006

a(n) = C(n,3) - C(n,1), n>=4. - Zerinvary Lajos, Nov 25 2006

a(n) = - A005581(-4-n) for all n in Z. - Michael Somos, Apr 13 2007

a(n) = A214292(n+4,2). - Reinhard Zumkeller, Jul 12 2012

From Amiram Eldar, Feb 20 2022: (Start)

Sum_{n>=1} 1/a(n) = 77/200.

Sum_{n>=1} (-1)^(n+1)/a(n) = 363/200 - 12*log(2)/5. (End)

a(n) = A005581(n+2)-2. - R. J. Mathar, Nov 22 2024

Extensions

M3842=A005555 in the 1995 EIS was the same sequence as this.

More terms from Zerinvary Lajos, Jan 09 2006