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%I #9 Oct 24 2022 15:12:44
%S 0,0,0,0,3,15,54,165,429,999,2145,4290,8100,14586,25194,41985,67830,
%T 106590,163431,245157,360525,520749,740025,1036035,1430703,1950975,
%U 2629575,3506085,4628052,6052068,7845255,10086780,12869340,16301142
%N The 3rd Witt transform of A000217.
%C The 2nd Witt transform of A000217 is essentially in A032092.
%H Vincenzo Librandi, <a href="/A147618/b147618.txt">Table of n, a(n) for n = 0..1000</a>
%H Pieter Moree, <a href="http://dx.doi.org/10.1016/j.disc.2005.03.004">The formal series Witt transform</a>, Discr. Math. no. 295 vol. 1-3 (2005) 143-160.
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,23,-33,51,-64,63,-63,64,-51,33,-23,15,-6,1).
%F G.f.: 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3).
%F a(n) = (1/81)*(n*(n+3)*(3*n^6 +27*n^5 +45*n^4 -135*n^3 -288*n^2 +108*n -2000)/4480 +2*A049347(n) +A049347(n-1) +(-1)^n*(A099254(n) -2*A099254(n- 1)) -3*(-1)^n*(A128504(n) -2*A128504(n-1))). - _G. C. Greubel_, Oct 24 2022
%t CoefficientList[Series[3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3), {x,0,40}], x] (* _Vincenzo Librandi_, Dec 13 2012 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); [0,0,0,0] cat Coefficients(R!( 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3) )); // _G. C. Greubel_, Oct 24 2022
%o (SageMath)
%o def A147618_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 3*x^4*(1-x+3*x^2-x^3+x^4)/((1-x)^9*(1+x+x^2)^3) ).list()
%o A147618_list(30) # _G. C. Greubel_, Oct 24 2022
%Y Cf. A000217, A032092, A049347, A099254, A128504.
%K easy,nonn
%O 0,5
%A _R. J. Mathar_, Nov 08 2008
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