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%I #37 Sep 08 2022 08:45:09
%S 0,1,4,15,56,205,732,2555,8752,29529,98420,324775,1062888,3454373,
%T 11160268,35872275,114791264,365897137,1162261476,3680494655,
%U 11622614680,36611236221,115063885244,360882185515,1129718145936
%N Expansion of e.g.f. x*exp(2*x)*cosh(x).
%C Binomial transform of A057711. 2nd binomial transform of (0,1,0,3,0,5,0,7,...).
%H G. C. Greubel, <a href="/A082133/b082133.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-22,24,-9).
%F a(n) = n*(1^(n-1) + 3^(n-1))/2.
%F E.g.f.: x*exp(2x)*cosh(x).
%F G.f.: x*(1-4*x+5*x^2) / ( (3*x-1)^2*(x-1)^2 ). - _R. J. Mathar_, Nov 24 2012
%F a(n) = Sum_{k=1..n} (Sum_{j=1..3} Stirling2(n,j)). - _G. C. Greubel_, Feb 07 2018
%p with (combinat):seq(sum(sum(stirling2(n, j),j=1..3), k=1..n), n=0..24); # _Zerinvary Lajos_, Dec 04 2007
%t With[{nn=30},CoefficientList[Series[x Exp[2x]Cosh[x],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Apr 30 2012 *)
%t Table[n*(1^(n-1) + 3^(n-1))/2, {n,0,30}] (* _G. C. Greubel_, Feb 05 2018 *)
%t Table[Sum[Sum[StirlingS2[n,j], {j,1,3}], {k,1,n}], {n,0,30}] (* _G. C. Greubel_, Feb 07 2018 *)
%o (PARI) for(n=0,30, print1(n*(1^(n-1) + 3^(n-1))/2, ", ")) \\ _G. C. Greubel_, Feb 05 2018
%o (Magma) [n*(1^(n-1) + 3^(n-1))/2: n in [0..30]]; // _G. C. Greubel_, Feb 05 2018
%o (GAP) List([0..10^2], n->Sum([1..n], k->Sum([1..3], j->Stirling2(n,j)))); # _Muniru A Asiru_, Feb 06 2018
%Y Cf. A082134, A082135, A082136.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Apr 06 2003
%E Definition clarified by _Harvey P. Dale_, Apr 30 2012
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