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 A098658 #61 Apr 24 2025 08:50:36
%S A098658 1,6,54,540,5670,61236,673596,7505784,84440070,956987460,10909657044,
%T A098658 124965162504,1437099368796,16581915793800,191876454185400,
%U A098658 2225766868550640,25874539846901190,301362287628613860
%N A098658 a(n) = 3^n*(2*n)!/(n!)^2.
%C A098658 Number of lattice paths from (0,0) to (n,n) using steps (0,1) and three kinds of steps (1,0). - _Joerg Arndt_, Jul 01 2011
%C A098658 Sixth binomial transform of 1/sqrt(1-36*x^2).
%C A098658 Diagonal of the rational function 1 / (1 - 3*x - y). - _Ilya Gutkovskiy_, Apr 24 2025
%H A098658 Vincenzo Librandi, <a href="/A098658/b098658.txt">Table of n, a(n) for n = 0..200</a>
%H A098658 Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%H A098658 Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%H A098658 Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F A098658 G.f.: 1/sqrt((1-6*x)^2-36*x^2) = 1/sqrt(1-12*x).
%F A098658 E.g.f.: exp(6*x)*BesselI(0, 6x).
%F A098658 a(n) = [t^n](1+6*t+9*t^2)^n.
%F A098658 a(n) = 3^n*A000984(n). - _R. J. Mathar_, Oct 10 2012
%F A098658 G.f.: Q(0), where Q(k) = 1 + 12*x*(4*k+1)/( 4*k+2 - 12*x*(4*k+2)*(4*k+3)/(12*x*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Sep 14 2013
%F A098658 n*a(n) +6*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Nov 27 2014
%F A098658 exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 6*x + 45*x^2 + 378*x^3 + ... is the o.g.f. for A101600. - _Peter Bala_, Jul 16 2015
%F A098658 From _Amiram Eldar_, Jul 21 2020: (Start)
%F A098658 Sum_{n>=0} 1/a(n) = 12/11 + 12*sqrt(11)*arcsin(1/sqrt(12))/121.
%F A098658 Sum_{n>=0} (-1)^n/a(n) = 12/13 - 12*sqrt(13)*arcsinh(1/sqrt(12))/169. (End)
%F A098658 From _Peter Bala_, Oct 12 2024: (Start)
%F A098658 a(n) = Integral_{x = 0..12} x^n * w(x) dx, where w(x) = 1/( Pi*sqrt(x*(12 - x)) ) is positive on the interval (0, 12). The weight function w(x) is singular at x = 0 and at x = 12 and is the solution of the Hausdorff moment problem.
%F A098658 Binomial transform of A322246.
%F A098658 The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r. (End)
%t A098658 Table[3^n (2n)!/(n!)^2,{n,0,20}] (* _Harvey P. Dale_, Dec 14 2011 *)
%o A098658 (PARI) /* same as in A092566 but use */
%o A098658 steps=[[1,0], [1,0], [1,0], [0,1]]; /* note the triple [1,0] */
%o A098658 /* _Joerg Arndt_, Jun 30 2011 */
%o A098658 (Magma) [3^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // _Vincenzo Librandi_, Jul 05 2011
%Y A098658 Cf. A000984, A059304, A101600, A322246.
%K A098658 easy,nonn
%O A098658 0,2
%A A098658 _Paul Barry_, Sep 20 2004