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%I #12 Sep 08 2022 08:45:56
%S 16,120,482,1412,3402,7168,13692,24264,40524,64504,98670,145964,
%T 209846,294336,404056,544272,720936,940728,1211098,1540308,1937474,
%U 2412608,2976660,3641560,4420260,5326776,6376230,7584892
%N Expansion of (16+24*x+2*x^2)/(x-1)^6.
%C Equals the sixth right hand column of A175136.
%H Vincenzo Librandi, <a href="/A190049/b190049.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F G.f.: (16 +24*x +2*x^2)/(1-x)^6.
%F a(n) = 16*binomial(n+5,5) +24*binomial(n+4,5) +2*binomial(n+3,5).
%F a(n) = A091894(5,0)*binomial(n+5,5) + A091894(5,1)*binomial(n+4,5) + A091894(5,2)*binomial(n+3,5).
%F a(n) = (21*n^5 +245*n^4 +1105*n^3 +2395*n^2 +2474*n +960)/60.
%p A190049 := proc(n) option remember; a(n):=(21*n^5 +245*n^4 +1105*n^3 +2395*n^2 +2474*n +960)/60 end: seq(A190049(n),n=0..27);
%t LinearRecurrence[{6,-15,20,-15,6,-1}, {16,120,482,1412,3402,7168}, 30] (* or *) CoefficientList[Series[(16 +24*x +2*x^2)/(1-x)^6, {x, 0, 50}], x] (* _G. C. Greubel_, Jan 10 2018 *)
%o (Magma) [(21*n^5+245*n^4+1105*n^3+2395*n^2+2474*n+960)/60: n in [0..50]]; // _Vincenzo Librandi_, May 07 2011
%o (PARI) x='x+O('x^30); Vec((16 +24*x +2*x^2)/(1-x)^6) \\ _G. C. Greubel_, Jan 10 2018
%o (PARI) for(n=0,30, print1((21*n^5 +245*n^4 +1105*n^3 +2395*n^2 +2474*n +960)/60, ", ")) \\ _G. C. Greubel_, Jan 10 2018
%Y Cf. A175136, A162148, A190048.
%Y Related to A000389 and A091894.
%K nonn,easy
%O 0,1
%A _Johannes W. Meijer_, May 06 2011
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