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 A383118 #15 Apr 17 2025 08:11:07
%S A383118 1,2,10,47,238,1232,6499,34715,187198,1016840,5555560,30497150,
%T A383118 168073195,929348396,5153362231,28646281502,159579236014,890644144580,
%U A383118 4979200476088,27878225498030,156298588113088,877350590047496,4930273302851830,27733610884176338
%N A383118 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(3*k,k).
%C A383118 Inverse binomial transform of A005809.
%F A383118 G.f.: (1/x) * Sum_{k>=0} binomial(3*k,k) * (x/(1 + x))^(k+1).
%F A383118 a(n) = [x^n] (1 + 2*x + 3*x^2 + x^3)^n.
%F A383118 The g.f. x * exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals Series_Reversion( x/((1+x)^3 - x) ). See A127897. - _Seiichi Manyama_, Apr 17 2025
%F A383118 a(n) ~ 23^(n + 1/2) / (3 * sqrt(Pi*n) * 2^(2*n+1)). - _Vaclav Kotesovec_, Apr 17 2025
%t A383118 Table[Sum[(-1)^(n - k) Binomial[n, k] Binomial[3 k, k], {k, 0, n}], {n, 0, 23}]
%t A383118 Table[(-1)^n HypergeometricPFQ[{1/3, 2/3, -n}, {1/2, 1}, 27/4], {n, 0, 23}]
%t A383118 nmax = 23; CoefficientList[Series[(1/x) Sum[Binomial[3 k, k] (x/(1 + x))^(k + 1), {k, 0, nmax}], {x, 0, nmax}], x]
%o A383118 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*binomial(3*k, k)); \\ _Seiichi Manyama_, Apr 17 2025
%Y A383118 Cf. A002426, A005809, A188686, A346628, A383119.
%Y A383118 Cf. A127897.
%K A383118 nonn
%O A383118 0,2
%A A383118 _Ilya Gutkovskiy_, Apr 17 2025