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%I #18 Sep 08 2022 08:45:32
%S 0,0,0,0,0,0,0,1,6,21,56,126,254,480,882,1617,2992,5580,10410,19292,
%T 35400,64343,116128,208701,374226,670095,1198164,2138423,3808148,
%U 6766089,11996042,21229790,37513896,66202347,116692472,205458357,361349662,634845141,1114205988
%N a(n) = Sum_{k=0..n/2} k*binomial(n-2*k, 3*k+2).
%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.
%H Alois P. Heinz, <a href="/A137361/b137361.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: x^7/(x^5 + x^3 - 3*x^2 + 3*x - 1)^2. - _Alois P. Heinz_, Oct 23 2008
%p a:= n-> (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)[1,8]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Oct 23 2008
%t t[i_, j_] := If[i == j-1, 1, If[j == 1, {6, -15, 20, -15, 8, -7, 6, -2, 0, -1}[[i]] , 0]]; M = Array[t, {10, 10}]; a[n_] := MatrixPower[M, n][[1, 8]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Feb 13 2015, after _Alois P. Heinz_ *)
%o (Magma) [&+[k*Binomial(n-2*k, 3*k+2): k in [0..(n div 2)]]: n in [0..40]]; // _Bruno Berselli_, Feb 13 2015
%Y Cf. A137356-A137360, A136444.
%K nonn
%O 0,9
%A _Don Knuth_, Apr 11 2008
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