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A194549 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

%I A194549 #25 Mar 05 2021 07:41:37
%S A194549 1,1,2,0,3,1,4,-1,2,1,5,0,3,2,6,-2,1,0,4,3,2,7,-1,2,1,5,0,4,3,8,-3,0,
%T A194549 -1,3,2,1,6,1,5,4,3,9,-2,1,0,4,-1,3,2,7,2,1,6,5,4,10,-4,-1,-2,2,1,0,5,
%U A194549 0,4,3,2,8,-1,3,2,7,1,6,5,4,11,-3,0,-1,3,-2
%N A194549 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.
%H A194549 Alois P. Heinz, <a href="/A194549/b194549.txt">Rows n = 1..33, flattened</a>
%F A194549 a(n) = A141285(n) - A194548(n).
%e A194549 Written as a triangle:
%e A194549   1;
%e A194549   1;
%e A194549   2;
%e A194549   0,3;
%e A194549   1,4;
%e A194549   -1,2,1,5;
%e A194549   0,3,2,6;
%e A194549   -2,1,0,4,3,2,7;
%e A194549   -1,2,1,5,0,4,3,8;
%e A194549   -3,0,-1,3,2,1,6,1,5,4,3,9;
%e A194549   -2,1,0,4,-1,3,2,7,2,1,6,5,4,10;
%e A194549   -4,-1,-2,2,1,0,5,0,4,3,2,8,-1,3,2,7,1,6,5,4,11;
%p A194549 T:= proc(n) local b, l;
%p A194549       b:= proc(n, i, t)
%p A194549             if n=0 then l:=l, i-t
%p A194549           elif i>n then
%p A194549           else b(n-i, i, t+1); b(n, i+1, t)
%p A194549             fi
%p A194549           end;
%p A194549       if n<2 then 1 else l:= NULL; b(n, 2, 0); l fi
%p A194549     end:
%p A194549 seq(T(n), n=1..13); # _Alois P. Heinz_, Dec 20 2011
%t A194549 T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
%t A194549      If[n0 == 0, l = Append[l, i-t],
%t A194549      If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
%t A194549      If[n<2, {1}, l = {}; b[n, 2, 0]; l]];
%t A194549 Table[T[n], {n, 1, 13}]  // Flatten (* _Jean-François Alcover_, Mar 05 2021, after _Alois P. Heinz_ *)
%Y A194549 The sum of row n is A000041(n-1). Row n has length A187219(n).
%Y A194549 Cf. A002865, A135010, A138121, A194546, A194547, A194548.
%K A194549 sign,tabf,look
%O A194549 1,3
%A A194549 _Omar E. Pol_, Dec 11 2011
%E A194549 More terms from _Alois P. Heinz_, Dec 20 2011