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 A082133 #40 Feb 21 2025 12:33:00
%S A082133 0,1,4,15,56,205,732,2555,8752,29529,98420,324775,1062888,3454373,
%T A082133 11160268,35872275,114791264,365897137,1162261476,3680494655,
%U A082133 11622614680,36611236221,115063885244,360882185515,1129718145936
%N A082133 Expansion of e.g.f. x*exp(2*x)*cosh(x).
%C A082133 Binomial transform of A057711. 2nd binomial transform of (0,1,0,3,0,5,0,7,...).
%H A082133 G. C. Greubel, <a href="/A082133/b082133.txt">Table of n, a(n) for n = 0..1000</a>
%H A082133 Mark Shattuck, <a href="https://arxiv.org/abs/2502.10661">Enumeration of consecutive patterns in flattened Catalan words</a>, arXiv:2502.10661 [math.CO], 2025. See pp. 3, 6.
%H A082133 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-22,24,-9).
%F A082133 a(n) = n*(1^(n-1) + 3^(n-1))/2.
%F A082133 E.g.f.: x*exp(2x)*cosh(x).
%F A082133 G.f.: x*(1-4*x+5*x^2) / ( (3*x-1)^2*(x-1)^2 ). - _R. J. Mathar_, Nov 24 2012
%F A082133 a(n) = Sum_{k=1..n} (Sum_{j=1..3} Stirling2(n,j)). - _G. C. Greubel_, Feb 07 2018
%p A082133 with (combinat):seq(sum(sum(stirling2(n, j),j=1..3), k=1..n), n=0..24); # _Zerinvary Lajos_, Dec 04 2007
%t A082133 With[{nn=30},CoefficientList[Series[x Exp[2x]Cosh[x],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Apr 30 2012 *)
%t A082133 Table[n*(1^(n-1) + 3^(n-1))/2, {n,0,30}] (* _G. C. Greubel_, Feb 05 2018 *)
%t A082133 Table[Sum[Sum[StirlingS2[n,j], {j,1,3}], {k,1,n}], {n,0,30}] (* _G. C. Greubel_, Feb 07 2018 *)
%o A082133 (PARI) for(n=0,30, print1(n*(1^(n-1) + 3^(n-1))/2, ", ")) \\ _G. C. Greubel_, Feb 05 2018
%o A082133 (Magma) [n*(1^(n-1) + 3^(n-1))/2: n in [0..30]]; // _G. C. Greubel_, Feb 05 2018
%o A082133 (GAP) List([0..10^2], n->Sum([1..n], k->Sum([1..3], j->Stirling2(n,j)))); # _Muniru A Asiru_, Feb 06 2018
%Y A082133 Cf. A082134, A082135, A082136.
%K A082133 easy,nonn
%O A082133 0,3
%A A082133 _Paul Barry_, Apr 06 2003
%E A082133 Definition clarified by _Harvey P. Dale_, Apr 30 2012