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A089274 Fifth column of the Legendre-Stirling triangle A071951.

%I A089274 #22 Nov 11 2024 02:04:38
%S A089274 1,70,3192,121424,4203824,137922336,4380918784,136378114048,
%T A089274 4191383868672,127754693361152,3873052857829376,117001609550671872,
%U A089274 3526270158211870720,106112798944292282368,3189880933574260359168
%N A089274 Fifth column of the Legendre-Stirling triangle A071951.
%C A089274 This is the fifth member of the family A000079 (powers of 2), A016129, A016309, A071952, etc.
%H A089274 Vincenzo Librandi, <a href="/A089274/b089274.txt">Table of n, a(n) for n = 0..500</a>
%H A089274 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (70,-1708,17544,-72000,86400).
%F A089274 G.f.: 1/((1-2*1*x)*(1-3*2*x)*(1-4*3*x)*(1-5*4*x)*(1-6*5*x)).
%F A089274 a(n) = (16875*(6*5)^n -20000*(5*4)^n +6048*(4*3)^n -405*(3*2)^n +2*(2*1)^n)/2520.
%F A089274 a(n) = A071951(n+5, 5), n>=0.
%F A089274 a(n) = det(|ps(i+5,j+4)|, 1 <= i,j <= n), where ps(n,k) are Legendre-Stirling numbers of the first kind (A129467). [_Mircea Merca_, Apr 06 2013]
%F A089274 E.g.f.: (1/2520)*(2*exp(2*x) - 405*exp(6*x) + 6048*exp(12*x) - 20000*exp(20*x) + 16875*exp(30*x)). - _G. C. Greubel_, Nov 10 2024
%t A089274 Table[2^(n-3)*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)/315, {n,0,30}] (* _G. C. Greubel_, Nov 10 2024 *)
%o A089274 (Magma) [(16875*(6*5)^n - 20000*(5*4)^n + 6048*(4*3)^n - 405*(3*2)^n + 2*(2*1)^n)/2520: n in [0..20]]; // _Vincenzo Librandi_, Sep 02 2011
%o A089274 (SageMath)
%o A089274 def A089274(n): return 2^n*(5*(15)^(n+3) -2*(10)^(n+4) +28*6^(n+3) -5*3^(n+4) +2)//2520
%o A089274 [A089274(n) for n in range(31)] # _G. C. Greubel_, Nov 10 2024
%Y A089274 Cf. A016129, A016309, A071951, A071952, A089278, A089500.
%Y A089274 Cf. A000079 (powers of 2).
%K A089274 nonn,easy
%O A089274 0,2
%A A089274 _Wolfdieter Lang_, Nov 07 2003