0,2

Number of walks on square lattice.

Number of standard tableaux of shape (n+2,3) (n >= 1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 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 (deutsch(AT)duke.poly.edu), Jan 25 2005

Column 4 of Catalan triangle A009766. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2006

A005586=Sum of first n Triangular numbers minus next Triangular number. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

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.

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].

S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6

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 (perry(AT)globalnet.co.uk), Jul 23 2003

C(5+n, 3)-C(5+n, 2) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 09 2006

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

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).

[seq(binomial(n, 3 )-binomial(n, 1), n=4..48)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 25 2006

a:=n->sum ((j-3)*j/2, j=0..n): seq(a(n), n=4..48); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 17 2006

A005586:=z*(5-6*z+2*z**2)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]

seq(sum(binomial(n, m), m=1..3)-n^2, n=5..49); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 19 2008

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 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]

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

Cf. A000217, A000292, A009766.

a(n)=A053121(n+5, n-1). A005581(n)= -a(-4-n).

Sequence in context: A073347 A134238 A024800 this_sequence A031333 A047801 A005918

Adjacent sequences: A005583 A005584 A005585 this_sequence A005587 A005588 A005589

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

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

More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 09 2006