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%I #13 Jan 27 2020 01:24:48
%S 1,2,1,1,4,1,1,6,6,1,1,6,15,8,1,1,7,23,28,10,1,1,8,30,60,45,12,1,1,9,
%T 39,98,125,66,14,1,1,10,49,144,255,226,91,16,1,1,11,60,202,437,561,
%U 371,120,18,1,1,12,72,272,685,1128,1092,568,153,20,1,1,13,85,355,1015,1995,2555
%N Triangle read by rows: T(n,k) is the number of compositions of n into k parts when there are two kinds of part 2.
%C Triangle T(n,k), 1 <= k <= n, given by (0, 2, -3/2, -1/6, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. Triangle T(n,k), 0 <= k <= n, is the Riordan array (1, x*(1+x-x^2)/(1-x)). - _Philippe Deléham_, Jan 25 2012
%F G.f. = t*z*(1+z-z^2)/(1-z-t*z-t*z^2+t*z^3).
%F T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-2j-1, k-j-1). - _Emeric Deutsch_, Aug 06 2006
%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) - T(n-3,k-1), n > 1. - _Philippe Deléham_, Jan 25 2012
%e T(4,2)=6 because we have (1,3),(3,1),(2,2),(2,2'),(2',2) and (2',2').
%e Triangle begins:
%e 1;
%e 2, 1;
%e 1, 4, 1;
%e 1, 6, 6, 1;
%e 1, 6, 15, 8, 1;
%e From _Philippe Deléham_, Jan 25 2012: (Start)
%e Triangle T(n,k) given by (0, 2, -3/2, -1/6, 2/3, 0, 0, 0, ...) DELTA (1,0,0,0,0,...) begins:
%e 1;
%e 0, 1;
%e 0, 2, 1;
%e 0, 1, 4, 1;
%e 0, 1, 6, 6, 1;
%e 0, 1, 6, 15, 8, 1; ... (End)
%p G:=t*z*(1+z-z^2)/(1-z-t*z-t*z^2+t*z^3): Gser:=simplify(series(G,z=0,15)): for n from 1 to 14 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 13 do seq(coeff(P[n],t^k),k=1..n) od; # yields sequence in triangular form
%Y Row sums yield A077998.
%Y Diagonals: A000012, A005843, A000384.
%K nonn,tabl
%O 1,2
%A _Emeric Deutsch_, Apr 09 2005
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