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 A382934 #10 Apr 17 2025 04:28:38
%S A382934 1,8,142,3188,79306,2091128,57251944,1609275536,46123258714,
%T A382934 1341870616928,39505611952852,1174352843125976,35189447673190864,
%U A382934 1061579548438995776,32210037668484980992,982173609216589910528,30079350892561552670554,924711257106480733093616,28524228913983070512002044
%N A382934 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(n+2*k,k) * 2^(n-k).
%C A382934 Diagonal of the rational function 1 / (1 - x - y - z - 2*x*y*z).
%F A382934 a(n) ~ sqrt(1/4 + sqrt(13)*cosh(arccosh(47/13^(3/2))/3)/6) * (1 + 2*cosh(arccosh(2)/3))^(3*n) / (Pi*n). Equivalently, a(n) ~ (1 + (2 - sqrt(3))^(1/3) + (2 + sqrt(3))^(1/3))^(3*n) / (sqrt(6*((2 - sqrt(3))^(1/3) + (2 + sqrt(3))^(1/3) - 2))*Pi*n). - _Vaclav Kotesovec_, Apr 17 2025
%t A382934 Table[Sum[Binomial[n, k] Binomial[n + k, k] Binomial[n + 2 k, k] 2^(n - k), {k, 0, n}], {n, 0, 18}]
%t A382934 Table[2^n HypergeometricPFQ[{n/2 + 1/2, n/2 + 1, -n}, {1, 1}, -2], {n, 0, 18}]
%t A382934 Table[SeriesCoefficient[1/(1 - x - y - z - 2 x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 18}]
%Y A382934 Cf. A001850, A069835, A081798, A126086.
%K A382934 nonn
%O A382934 0,2
%A A382934 _Ilya Gutkovskiy_, Apr 09 2025