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A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

%I A194548 #31 Mar 05 2021 07:49:17
%S A194548 0,1,1,2,1,2,1,3,2,2,1,3,2,2,1,4,3,3,2,2,2,1,4,3,3,2,3,2,2,1,5,4,4,3,
%T A194548 3,3,2,3,2,2,2,1,5,4,4,3,4,3,3,2,3,3,2,2,2,1,6,5,5,4,4,4,3,4,3,3,3,2,
%U A194548 4,3,3,2,3,2,2,2,1,6,5,5,4,5,4,4,3,4,4,3,3,3,2,4,3,3,3,2,3,2,2,2,1
%N A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.
%H A194548 Alois P. Heinz, <a href="/A194548/b194548.txt">Rows n = 1..33, flattened</a>
%H A194548 Tilman Piesk, <a href="/A194602/a194602.txt">Table</a> for A194602, showing the non-one addends.
%e A194548 Written as a triangle:
%e A194548   0;
%e A194548   1;
%e A194548   1;
%e A194548   2,1;
%e A194548   2,1;
%e A194548   3,2,2,1;
%e A194548   3,2,2,1;
%e A194548   4,3,3,2,2,2,1;
%e A194548   4,3,3,2,3,2,2,1;
%e A194548   5,4,4,3,3,3,2,3,2,2,2,1;
%e A194548   5,4,4,3,4,3,3,2,3,3,2,2,2,1;
%e A194548   6,5,5,4,4,4,3,4,3,3,3,2,4,3,3,2,3,2,2,2,1;
%e A194548   6,5,5,4,5,4,4,3,4,4,3,3,3,2,4,3,3,3,2,3,2,2,2,1;
%p A194548 T:= proc(n) local b, l;
%p A194548       b:= proc(n, i, t)
%p A194548             if n=0 then l:=l, t
%p A194548           elif i>n then
%p A194548           else b(n-i, i, t+1); b(n, i+1, t)
%p A194548             fi
%p A194548           end;
%p A194548       if n<2 then 0 else l:= NULL; b(n, 2, 0); l fi
%p A194548     end:
%p A194548 seq(T(n), n=1..15);  # _Alois P. Heinz_, Dec 19 2011
%t A194548 T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
%t A194548      If[n0==0, l = Append[l, t],
%t A194548      If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
%t A194548      If[n<2, {0}, l = {}; b[n, 2, 0]; l]];
%t A194548 Table[T[n], {n, 1, 15}]  // Flatten (* _Jean-François Alcover_, Mar 05 2021, after _Alois P. Heinz_ *)
%Y A194548 Row sums give A138135. Row n has length A187219(n).
%Y A194548 Cf. A002865, A135010, A138121, A193173, A194546, A194547, A194549.
%K A194548 nonn,tabf
%O A194548 1,4
%A A194548 _Omar E. Pol_, Dec 11 2011
%E A194548 More terms from _Alois P. Heinz_, Dec 19 2011