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%I M3841 #81 Feb 20 2022 22:37:11
%S 0,5,14,28,48,75,110,154,208,273,350,440,544,663,798,950,1120,1309,
%T 1518,1748,2000,2275,2574,2898,3248,3625,4030,4464,4928,5423,5950,
%U 6510,7104,7733,8398,9100,9840,10619,11438,12298,13200,14145,15134,16168,17248
%N a(n) = n*(n+4)*(n+5)/6.
%C Number of walks on square lattice.
%C Number of standard tableaux of shape (n+2,3) (n >= 1). - _Emeric Deutsch_, May 20 2004
%C 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
%C Column 4 of Catalan triangle A009766. - _Zerinvary Lajos_, Nov 25 2006
%C Sum of first n triangular numbers minus next triangular number. - _Vladimir Joseph Stephan Orlovsky_, Oct 13 2009
%C Number of packed increasing tableaux of shape 3 X (n+1) with alphabet [n+4]. - _Oliver Pechenik_, Jan 03 2022
%D 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A005586/b005586.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H C. Gaetz, O. Pechenik, J. Striker, and J. P. Swanson, <a href="https://arxiv.org/abs/2112.09228">Curious cyclic sieving on increasing tableaux</a>, arXiv:2112.09228 [math.CO], 2021. See Proposition 1.1 at the top of page 2.
%H Richard K. Guy, <a href="/A005581/a005581_1.pdf">Letter to N. J. A. Sloane, Feb 1988</a>
%H Richard K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, Sandsteps and Pascal Pyramids</a>, J. Integer Seqs., Vol. 3 (2000), Article 00.1.6.
%H Ângela Mestre and José Agapito, <a href="https://www.emis.de/journals/JIS/VOL22/Agapito/mestre8.html">A Family of Riordan Group Automorphisms</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992, arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: x * (5 - 6*x + 2*x^2) / (1 - x)^4.
%F E.g.f.: (5*x + 2*x^2 + x^3/6) * exp(x). - _Michael Somos_, Apr 13 2007
%F 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
%F a(n) = C(5+n, 3)-C(5+n, 2). - _Zerinvary Lajos_, Jan 09 2006
%F a(n) = C(n,3) - C(n,1), n>=4. - _Zerinvary Lajos_, Nov 25 2006
%F a(n) = - A005581(-4-n) for all n in Z. - _Michael Somos_, Apr 13 2007
%F a(n) = A214292(n+4,2). - _Reinhard Zumkeller_, Jul 12 2012
%F From _Amiram Eldar_, Feb 20 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 77/200.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 363/200 - 12*log(2)/5. (End)
%e G.f. = 5*x + 14*x^2 + 28*x^3 + 48*x^4 + 75*x^5 + 110*x^6 + 154*x^7 + ...
%p [seq(binomial(n,3 )-binomial(n,1),n=4..48)]; # _Zerinvary Lajos_, Nov 25 2006
%p a:=n->sum ((j-3)*j/2,j=0..n): seq(a(n),n=4..48); # _Zerinvary Lajos_, Dec 17 2006
%p A005586:=z*(5-6*z+2*z**2)/(z-1)**4; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p seq(sum(binomial(n,m), m=1..3)-n^2,n=5..49); # _Zerinvary Lajos_, Jun 19 2008
%t 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 *)
%t CoefficientList[Series[x (5 - 6 x + 2 x^2) / (1 - x)^4, {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 09 2013 *)
%t 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 *)
%o (PARI) {a(n) = n * (n+4) * (n+5) / 6}; /* _Michael Somos_, Apr 13 2007 */
%o (Magma) [n*(n+4)*(n+5)/6: n in [0..50]]; // _Vincenzo Librandi_, Jun 09 2013
%Y Cf. A000217, A000292, A005581, A009766, A053121.
%Y a(n)=A053121(n+5,n-1).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E M3842=A005555 in the 1995 EIS was the same sequence as this.
%E More terms from _Zerinvary Lajos_, Jan 09 2006
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