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A190051 Expansion of (1-x)*(10*x^4-20*x^3+16*x^2-6*x+1)/(1-2*x)^5.

%I A190051 #20 Jun 02 2025 04:04:44
%S A190051 1,3,12,44,150,482,1476,4344,12368,34240,92544,244992,636928,1629696,
%T A190051 4111360,10242048,25227264,61505536,148570112,355860480,845807616,
%U A190051 1996095488,4680056832,10906763264,25275924480,58271465472
%N A190051 Expansion of (1-x)*(10*x^4-20*x^3+16*x^2-6*x+1)/(1-2*x)^5.
%C A190051 The third left hand column of triangle A175136.
%H A190051 G. C. Greubel, <a href="/A190051/b190051.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..300 from Vincenzo Librandi)
%H A190051 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (10,-40,80,-80,32).
%F A190051 G.f.: (1-x)*(10*x^4-20*x^3+16*x^2-6*x+1)/(1-2*x)^5.
%F A190051 a(n) = (264 + 214*n + 14*n^3 + 83*n^2 + n^4)*2^(n-7)/3 for n >=1 with a(0)=1.
%F A190051 a(n-4) = A003472(n) -7*A003472(n-1) +22*A003472(n-2) -36*A003472(n-3) +30*A003472(n-4) -10*A003472(n-5) for n>=5 with a(0) = 1.
%p A190051 A190051:= proc(n) option remember; if n=0 then A190051(n):=1 else A190051(n):= (264+214*n+14*n^3+83*n^2+n^4)*2^(n-7)/3 fi: end: seq (A190051(n), n=0..25);
%t A190051 Join[{1}, LinearRecurrence[{10,-40,80,-80,32}, {3,12,44,150,482}, 30]] (* or *) CoefficientList[Series[(1 - x)*(10*x^4 -20*x^3 +16*x^2 -6*x + 1)/(1 -2*x)^5, {x, 0, 50}], x] (* _G. C. Greubel_, Jan 10 2018 *)
%o A190051 (PARI) x='x+O('x^30); Vec((1-x)*(10*x^4-20*x^3+16*x^2-6*x+1)/(1-2*x)^5) \\ _G. C. Greubel_, Jan 10 2018
%o A190051 (PARI) for(n=0,30, print1(if(n==0,1,(264 + 214*n + 14*n^3 + 83*n^2 + n^4)*2^(n-7)/3), ", ")) \\ _G. C. Greubel_, Jan 10 2018
%o A190051 (Magma) [1] cat [(264 + 214*n + 14*n^3 + 83*n^2 + n^4)*2^(n-7)/3: n in [1..30]]; // _G. C. Greubel_, Jan 10 2018
%Y A190051 Cf. A175136, A011782, A190050.
%Y A190051 Related to A003472.
%K A190051 nonn,easy
%O A190051 0,2
%A A190051 _Johannes W. Meijer_, May 06 2011