This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A116411 #41 Jun 25 2020 18:32:22
%S A116411 1,1,1,4,13,31,76,211,589,1597,4351,12046,33496,93094,259351,725089,
%T A116411 2031709,5701189,16023181,45104044,127134283,358764613,1013494318,
%U A116411 2865933907,8111573416,22977551656,65138143006,184789086106,524571000799,1490044262503,4234901256781,12042611698876,34262320521661,97525522430989
%N A116411 Coefficient of x^n in the expansion of (1+x+x^3)^n.
%C A116411 Binomial transform of central coefficient of (1+x^3)^n.
%C A116411 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (1,1), (1,3). [_Joerg Arndt_, Jul 05 2011]
%C A116411 a(n) also gives the number of lattice paths from (0,0) to (n,2*n) using steps (1,0), (1,2) and (1,3) since if a*(1,0) + b*(1,1) + c*(1,3) denotes a path from (0,0) to (n,n) then c*(1,0) + b*(1,2) + a*(1,3) is a path from (0,0) to (n,2*n) and vice versa. - _Peter Bala_, Jul 20 2014
%H A116411 Seiichi Manyama, <a href="/A116411/b116411.txt">Table of n, a(n) for n = 0..1000</a>
%H A116411 Hacène Belbachir, Abdelghani Mehdaoui, <a href="https://lrecits.usthb.dz/6.5.pdf">Diagonal sums in Pascal pyramid (1, 2, r)</a>, Les Annales RECITS (2019) Vol. 6, 45-52.
%F A116411 a(n) = Sum_{k=0..floor(n/3)} C(n, 3*k)*C(3*k, k).
%F A116411 a(n) = Sum_{k=0..n} C(n, k)*C(k, k/3)*(cos(2*Pi*k/3)+1/2)*2/3.
%F A116411 G.f.: A(x) satisfies (31*x^3-12*x^2+12*x-4)*A(x)^3+(3-3*x)*A(x)+1 = 0. - _Mark van Hoeij_, Apr 16 2013
%F A116411 From _Peter Bala_, Jul 20 2014: (Start)
%F A116411 a(n) = coefficient of x^(2*n) in the expansion of (1 + x^2 + x^3)^n.
%F A116411 O.g.f.: sum {n >= 0} binomial(3*n,n)*x^(3*n)/(1 - x)^(3*n + 1).
%F A116411 By the Lagrange inversion formula, the o.g.f. equals the logarithmic derivative of the o.g.f. of A071879.
%F A116411 E.g.f.: exp(x)*sum {n >= 0} x^(3*n)/((2*n)!*n!). (End)
%F A116411 Recurrence: 2*n*(2*n-3)*a(n) = 2*(6*n^2 - 12*n + 5)*a(n-1) - 6*(n-1)*(2*n-3)*a(n-2) + 31*(n-2)*(n-1)*a(n-3). - _Vaclav Kotesovec_, Feb 11 2015
%F A116411 a(n) ~ (3 + 2^(2/3))^(n+1/2) / (2^(2*n/3 + 1) * sqrt(3*Pi*n)). - _Vaclav Kotesovec_, Feb 11 2015
%F A116411 a(n) = 3F2[1/3-n/3,2/3-n/3,-n/3; 1/2,1; -27/4]. - _Benedict W. J. Irwin_, Nov 01 2016
%p A116411 series(RootOf( (31*x^3-12*x^2+12*x-4)*A^3+(3-3*x)*A+1, A),x=0,30); # _Mark van Hoeij_, Apr 16 2013
%t A116411 Flatten[{1,Table[Coefficient[Expand[(1+x+x^3)^n],x^n],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 11 2015 *)
%t A116411 Table[HypergeometricPFQ[{1/3-n/3,2/3-n/3,-n/3},{1/2,1},-27/4],{n,0,20}] (* _Benedict W. J. Irwin_, Nov 01 2016 *)
%o A116411 (PARI) a(n)=polcoeff((1+x+x^3)^n,n); /* _Joerg Arndt_, Jul 01 2011 */
%Y A116411 Cf. A071879.
%K A116411 easy,nonn
%O A116411 0,4
%A A116411 _Paul Barry_, Feb 13 2006