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%I #33 Sep 08 2022 08:44:52
%S 0,1,9,45,173,573,1725,4861,13053,33789,84989,208893,503805,1196029,
%T 2801661,6488061,14876669,33816573,76283901,170917885,380633085,
%U 843055101,1858076669,4076863485,8908701693,19394461693,42077257725,90999619581,196226318333
%N a(n) = A036800(n)/2.
%C Binomial transform of A054569 (with leading 0). Partial sums of A014477 (with leading 0). - _Paul Barry_, Jun 11 2003
%C This sequence is related to A000337 by a(n) = n*A000337(n) - Sum_{i=0..n-1} A000337(i). - _Bruno Berselli_, Mar 06 2012
%H Bruno Berselli, <a href="/A036826/b036826.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (7,-18,20,-8).
%F From _Paul Barry_, Jun 11 2003: (Start)
%F G.f.: x*(1+2*x)/((1-x)*(1-2*x)^3).
%F a(n) = 2^n*(n^2-2*n+3) - 3.
%F a(n) = Sum_{k=0..n} k^2*2^(k-1). (End)
%F a(n) = 7*a(n-1) -18*a(n-2) +20*a(n-3) -8*a(n-4). - _Harvey P. Dale_, Mar 04 2015
%F E.g.f.: -3*exp(x) + (3 -2*x +4*x^2)*exp(2*x). - _G. C. Greubel_, Mar 31 2021
%p A036826:= n-> 2^n*(3-2*n+n^2) -3; seq(A036826(n), n=0..30); # _G. C. Greubel_, Mar 31 2021
%t LinearRecurrence[{7,-18,20,-8}, {0,1,9,45}, 29] (* _Bruno Berselli_, Mar 06 2012 *)
%o (Magma) m:=28; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((1+2*x)/((1-x)*(1-2*x)^3))); // _Bruno Berselli_, Mar 06 2012
%o (PARI) for(n=0, 28, print1(2^n*(n^2-2*n+3)-3", ")); \\ _Bruno Berselli_, Mar 06 2012
%o (Sage) [2^n*(3-2*n+n^2) -3 for n in (0..30)] # _G. C. Greubel_, Mar 31 2021
%Y Cf. A000337, A014477, A036800, A054569, A209359.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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