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%I #33 Jan 11 2024 01:46:49
%S 1,13,105,675,3780,19278,91854,415530,1804275,7577955,30961359,
%T 123589557,483611310,1860043500,7046907660,26344593252,97328636181,
%U 355781149065,1288173125925,4623863536215,16466920464456,58222325927898,204499905118650,713919106104750
%N a(n) = (n+1)*(n+2)*(n+3)*(n+12)*3^n/72.
%H Harvey P. Dale, <a href="/A080422/b080422.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (15,-90,270,-405,243).
%F G.f.: (1-2*x)/(1-3*x)^5.
%F For n>4, a(n)=15*a(n-1)-90*a(n-2)+270*a(n-3)-405*a(n-4)+243*a(n-5). - _Harvey P. Dale_, Oct 22 2011
%F From _G. C. Greubel_, Dec 22 2023: (Start)
%F a(n) = A136158(n+4,4).
%F E.g.f.: (1/8)*(8 + 80*x + 144*x^2 + 72*x^3 + 9*x^4)*exp(3*x). (End)
%F From _Amiram Eldar_, Jan 11 2024: (Start)
%F Sum_{n>=0} 1/a(n) = 662816499/42350 - 2122848*log(3/2)/55.
%F Sum_{n>=0} (-1)^n/a(n) = 2135808*log(4/3)/55 - 94614897/8470. (End)
%t Table[(n + 1) (n + 2) (n + 3) (n + 12) 3^n/72, {n, 0, 30}] (* or *) LinearRecurrence[ {15, -90, 270, -405, 243}, {1, 13, 105, 675, 3780}, 30] (* _Harvey P. Dale_, Oct 22 2011 *)
%t CoefficientList[Series[(1 - 2 x) / (1 - 3 x)^5, {x, 0, 30}], x] (* _Vincenzo Librandi_, Aug 05 2013 *)
%o (Magma) [(n+1)*(n+2)*(n+3)*(n+12)*3^n/72: n in [0..30]]; // _Vincenzo Librandi_, Aug 05 2013
%o (SageMath) [(n+1)*(n+2)*(n+3)*(n+12)*3^(n-3)/8 for n in range(31)] # _G. C. Greubel_, Dec 22 2023
%Y T(n, 4) in triangle A080419.
%Y Cf. A136158.
%K nonn,easy
%O 0,2
%A _Paul Barry_, Feb 19 2003
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