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A005586 a(n) = n*(n+4)*(n+5)/6.
(Formerly M3841)
20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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).

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)

EXAMPLE

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

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 *)

PROG

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

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

CROSSREFS

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

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

Sequence in context: A073347 A134238 A024800 * A197058 A212678 A244100

Adjacent sequences: A005583 A005584 A005585 * A005587 A005588 A005589

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

More terms from Zerinvary Lajos, Jan 09 2006

STATUS

approved

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Last modified December 7 22:02 EST 2022. Contains 358671 sequences. (Running on oeis4.)