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A128235 Triangle read by rows: T(n,k) is the number of sequences of length n on the alphabet {0,1,2,3}, containing k subsequences 00 (0<=k<=n-1).

%I A128235 #6 Dec 12 2013 19:28:21
%S A128235 1,4,15,1,57,6,1,216,33,6,1,819,162,36,6,1,3105,756,189,39,6,1,11772,
%T A128235 3402,945,216,42,6,1,44631,14931,4536,1143,243,45,6,1,169209,64314,
%U A128235 21168,5778,1350,270,48,6,1,641520,273051,96633,28323,7128,1566,297,51
%N A128235 Triangle read by rows: T(n,k) is the number of sequences of length n on the alphabet {0,1,2,3}, containing k subsequences 00 (0<=k<=n-1).
%C A128235 Row n has n terms (n>=1). T(n,0) = A125145(n). Sum(k*T(n,k), k=0..n-1) = (n-1)*4^(n-2) = A002697(n-1).
%H A128235 Alois P. Heinz, <a href="/A128235/b128235.txt">Rows n = 0..150, flattened</a>
%F A128235 G.f.: (1+z-tz)/(1-3z-3z^2-tz+3tz^2).
%e A128235 T(4,2) = 6 because we have 0001, 0002, 0003, 1000, 2000 and 3000.
%e A128235 Triangle starts:
%e A128235 1;
%e A128235 4;
%e A128235 15,    1;
%e A128235 57,    6,  1;
%e A128235 216,  33,  6, 1;
%e A128235 819, 162, 36, 6, 1;
%p A128235 G:=(1+z-t*z)/(1-3*z-3*z^2-t*z+3*t*z^2): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form
%Y A128235 Cf. A125145, A002697.
%K A128235 nonn,tabf
%O A128235 0,2
%A A128235 _Emeric Deutsch_, Feb 27 2007