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A232396 Triangular array read by rows: T(n,k) is the number of compositions of n with no two consecutive identical parts that have exactly k parts = 1, n>=0, 0<=k<=ceiling(n/3).

%I A232396 #14 Nov 24 2013 14:18:44
%S A232396 1,0,1,1,0,1,2,1,2,1,3,3,1,3,8,3,6,9,7,1,7,20,10,2,14,27,25,5,18,52,
%T A232396 39,14,1,30,77,78,26,3,45,132,133,60,8,66,213,240,117,24,1,107,334,
%U A232396 421,232,54,4,157,562,716,450,127,12,245,872,1265,842,279,38,1
%N A232396 Triangular array read by rows: T(n,k) is the number of compositions of n with no two consecutive identical parts that have exactly k parts = 1, n>=0, 0<=k<=ceiling(n/3).
%H A232396 Alois P. Heinz, <a href="/A232396/b232396.txt">Rows n = 0..250, flattened</a>
%F A232396 G.f.: 1/( 1 - y*x/(1 + y*x) - Sum_{j>=2} x^j/(1 + x^j) ).
%e A232396 1;
%e A232396 0,  1;
%e A232396 1,  0;
%e A232396 1,  2;
%e A232396 1,  2,  1;
%e A232396 3,  3,  1;
%e A232396 3,  8,  3;
%e A232396 6,  9,  7,  1;
%e A232396 7,  20, 10, 2;
%e A232396 14, 27, 25, 5;
%e A232396 18, 52, 39, 14, 1;
%e A232396 T(7,2) = 7 because we have: 1+2+1+3, 1+2+3+1, 1+3+1+2, 1+3+2+1, 1+5+1, 2+1+3+1, 3+1+2+1.
%p A232396 b:= proc(n, t) option remember; `if`(n=0, 1, expand(
%p A232396       add(`if`(j=t, 0, b(n-j, j)*`if`(j=1, x, 1)), j=1..n)))
%p A232396     end:
%p A232396 T:= n-> seq(coeff(b(n, 0), x, i), i=0..ceil(n/3)):
%p A232396 seq(T(n), n=0..20); # _Alois P. Heinz_, Nov 24 2013
%t A232396 nn=10;CoefficientList[Series[1/(1- u z/(1+ u z) - Sum[z^j/(1+z^j),{j,2,nn}]),{z,0,nn}],{z,u}]//Grid
%Y A232396 Row sums give: A003242.
%K A232396 nonn,tabf
%O A232396 0,7
%A A232396 _Geoffrey Critzer_, Nov 23 2013