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 A236194 #46 Mar 12 2024 23:49:19
%S A236194 1,7,45,286,1820,11628,74613,480700,3108105,20160075,131128140,
%T A236194 854992152,5586853480,36576848168,239877544005,1575580702584,
%U A236194 10363194502115,68248282427325,449972009097765,2969831763694950,19619725782651120,129728497393775280
%N A236194 a(n) = binomial(3n+1, n-1).
%C A236194 This sequence is related to A006013 by a(n)/n = A006013(n)/2.
%H A236194 Bruno Berselli, <a href="/A236194/b236194.txt">Table of n, a(n) for n = 1..100</a>
%H A236194 Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%F A236194 G.f.: (sqrt(4-27*x)*cos((2/3)*arcsin((3/2)*sqrt(3*x))) + sqrt(3*x)*sin((2/3)*arcsin((3/2)*sqrt(3*x))) - sqrt(4-27*x))/(3*sqrt(4-27*x)*x^2). - _Emanuele Munarini_, Oct 14 2014
%F A236194 From _Peter Bala_, Nov 04 2015: (Start)
%F A236194 With offset 0, the o.g.f. equals f(x)*g(x)^4, where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764.
%F A236194 More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A013698 (k = 5), A165817 (k = -1), A117671 (k = -2). (End)
%F A236194 a(n) = [x^n] x/(1 - x)^(2*n+3). - _Ilya Gutkovskiy_, Oct 10 2017
%F A236194 From _Karol A. Penson_, Mar 02 2024: (Start)
%F A236194 G.f.: ((sqrt(3)*sqrt(x)*i + sqrt(4 - 27*x))*(4*sqrt(4 - 27*x) - 12*i*sqrt(3)*sqrt(x))^(2/3) + (-sqrt(3)*sqrt(x)*i + sqrt(4 - 27*x))*(4*sqrt(4 - 27*x) + 12*i*sqrt(3)*sqrt(x))^(2/3) - 8*sqrt(4 - 27*x))/(24*sqrt(4 - 27*x)*x^2), where i is the imaginary unit, i=sqrt(-1).
%F A236194 G.f.: hypergeometric3F2([5/3,2,7/3],[5/2,3],27*x/4).
%F A236194 G.f. = G satisfies the algebraic equation: 1 + (7*z-1)*G + (27*z-4)*z^2*G^2 + (27*z-4)*z^4*G^3 = 0. (End)
%t A236194 Table[Binomial[3n+1, n-1], {n, 30}]
%o A236194 (Magma) [Binomial(3*n+1,n-1): n in [1..30]];
%o A236194 (Maxima) makelist(binomial(3*n+4,n),n,0,40); /* _Emanuele Munarini_, Oct 14 2014 */
%o A236194 (PARI) vector(30, n, binomial(3*n+1, n-1)) \\ _Altug Alkan_, Nov 04 2015
%o A236194 (SageMath) [binomial(3*n+1,n-1) for n in range(1,31)] # _G. C. Greubel_, Nov 09 2022
%Y A236194 Cf. A006013; A025174: C(3n-1, n-1); A117671: C(3n+1, n+1).
%Y A236194 Second column of the triangle A159841.
%Y A236194 Third column of the triangle A119301.
%Y A236194 Cf. A001764, A004319, A005809, A013698, A045721, A165817.
%Y A236194 Cf. A045721, A117671.
%K A236194 nonn,easy
%O A236194 1,2
%A A236194 _Bruno Berselli_, Jan 20 2014