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A055232 Expansion of (1+2*x+3*x^2)/((1-x)^3*(1-x^2)).

%I A055232 #39 Sep 08 2022 08:45:01
%S A055232 1,5,16,36,69,117,184,272,385,525,696,900,1141,1421,1744,2112,2529,
%T A055232 2997,3520,4100,4741,5445,6216,7056,7969,8957,10024,11172,12405,13725,
%U A055232 15136,16640,18241,19941,21744,23652,25669,27797,30040,32400,34881,37485,40216,43076
%N A055232 Expansion of (1+2*x+3*x^2)/((1-x)^3*(1-x^2)).
%C A055232 a(n) is the number of (w,x,y) having all terms in {0..n} and w <= floor((x+y)/2). - _Clark Kimberling_, Jun 02 2012
%C A055232 First differences are in A212959. - _Wesley Ivan Hurt_, Apr 16 2016
%D A055232 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.28(c), y_3.
%H A055232 Michel Marcus, <a href="/A055232/b055232.txt">Table of n, a(n) for n = 0..999</a>
%H A055232 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A055232 G.f.: (1+2*x+3*x^2)/((1-x)^3*(1-x^2)).
%F A055232 a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5). - _Clark Kimberling_, Jun 02 2012
%F A055232 a(n) = (7+(-1)^n+16*n+14*n^2+4*n^3)/8. - _Eric W. Weisstein_, Apr 04 2013
%F A055232 a(n) = ((n+1)^3 + ceiling((n+1)/2)^2 + floor((n+1)/2)^2)/2. - _Wesley Ivan Hurt_, Apr 15 2016
%F A055232 E.g.f.: ((7 + 34*x + 26*x^2 + 4*x^3)*exp(x) + exp(-x))/8. - _Ilya Gutkovskiy_, Apr 16 2016
%p A055232 A055232:=n->(7+(-1)^n+16*n+14*n^2+4*n^3)/8: seq(A055232(n), n=0..100); # _Wesley Ivan Hurt_, Apr 15 2016
%t A055232 Table[(7 + (-1)^n + 16*n + 14*n^2 + 4*n^3)/8, {n, 0, 50}] (* _Wesley Ivan Hurt_, Apr 15 2016 *)
%t A055232 LinearRecurrence[{3,-2,-2,3,-1},{1,5,16,36,69},40] (* _Harvey P. Dale_, Oct 25 2020 *)
%o A055232 (Magma) [(7+(-1)^n+16*n+14*n^2+4*n^3)/8 : n in [0..100]]; // _Wesley Ivan Hurt_, Apr 15 2016
%o A055232 (PARI) lista(nn) = for(n=0, nn, print1((7+(-1)^n+16*n+14*n^2+4*n^3)/8, ", ")); \\ _Altug Alkan_, Apr 16 2016
%Y A055232 Cf. A212959.
%K A055232 nonn,easy
%O A055232 0,2
%A A055232 _N. J. A. Sloane_, Jul 05 2000