%I A005586 M3841
%S A005586 0,5,14,28,48,75,110,154,208,273,350,440,544,663,798,950,1120,1309,1518,
%T A005586 1748,2000,2275,2574,2898,3248,3625,4030,4464,4928,5423,5950,6510,7104,
%U A005586 7733,8398,9100,9840,10619,11438,12298,13200,14145,15134,16168,17248
%N A005586 a(n) = n(n+4)(n+5)/6.
%C A005586 Number of walks on square lattice.
%C A005586 Number of standard tableaux of shape (n+2,3) (n >= 1). - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), May 20 2004
%C A005586 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
%C A005586 Column 4 of Catalan triangle A009766. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 25 2006
%C A005586 A005586=Sum of first n Triangular numbers minus next Triangular number. 
               [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 13 2009]
%D A005586 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005586 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.
%D A005586 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.
%H A005586 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A005586 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005586 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A005586 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">J. Integer Seqs., Vol. 3 (2000), #00.1.6</
               a>
%F A005586 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
%F A005586 C(5+n, 3)-C(5+n, 2) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jan 09 2006
%F A005586 a(n)=C(n,3 )-C(n,1),n>=4 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 25 2006
%F A005586 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).
%p A005586 [seq(binomial(n,3 )-binomial(n,1),n=4..48)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 25 2006
%p A005586 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
%p A005586 A005586:=z*(5-6*z+2*z**2)/(z-1)**4; [Conjectured by S. Plouffe in his 
               1992 dissertation.]
%p A005586 seq(sum(binomial(n,m), m=1..3)-n^2,n=5..49); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 19 2008
%t A005586 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]
%o A005586 (PARI) {a(n)= n* (n+4)* (n+5)/6} /* MIchael Somos Apr 13 2007 */
%Y A005586 Cf. A000217, A000292, A009766.
%Y A005586 a(n)=A053121(n+5, n-1). A005581(n)= -a(-4-n).
%Y A005586 Sequence in context: A073347 A134238 A024800 this_sequence A031333 A047801 
               A005918
%Y A005586 Adjacent sequences: A005583 A005584 A005585 this_sequence A005587 A005588 
               A005589
%K A005586 nonn,easy
%O A005586 0,2
%A A005586 N. J. A. Sloane (njas(AT)research.att.com).
%E A005586 M3842=A005555 in the 1995 EIS was the same sequence as this.
%E A005586 More terms from Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 09 
               2006