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a(n) = Sum_{k <= n/2 } k*binomial(n-2k, 3k+1).
4

%I #12 May 31 2017 10:33:20

%S 0,0,0,0,0,0,1,5,15,35,70,128,226,402,735,1375,2588,4830,8882,16108,

%T 28943,51785,92573,165525,295869,528069,940259,1669725,2957941,

%U 5229953,9233748,16284106,28688451,50490125,88765885,155891305,273495479,479360847,839451764

%N a(n) = Sum_{k <= n/2 } k*binomial(n-2k, 3k+1).

%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (6, -15, 20, -15, 8, -7, 6, -2, 0, -1).

%F G.f.: x^6*(1-x)/(x^5+x^3-3*x^2+3*x-1)^2. - _Alois P. Heinz_, Oct 23 2008

%p a:= n-> (Matrix([[35, 15, 5, 1, 0$6]]). Matrix (10, (i,j)-> if i=j-1 then 1 elif j=1 then [6, -15, 20, -15, 8, -7, 6, -2, 0, -1][i] else 0 fi)^n)[1,10]: seq (a(n), n=0..50); # _Alois P. Heinz_, Oct 23 2008

%t Table[Sum[k Binomial[n-2k,3k+1],{k,n/2}],{n,0,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,8,-7,6,-2,0,-1},{0,0,0,0,0,0,1,5,15,35},40] (* _Harvey P. Dale_, May 31 2017 *)

%Y Cf. A137356-A137361, A136444.

%K nonn

%O 0,8

%A _Don Knuth_, Apr 11 2008