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A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.

%I A196930 #24 Jan 30 2014 09:21:42
%S A196930 1,2,3,2,4,2,5,2,2,3,6,2,2,3,7,2,2,2,2,3,4,8,2,2,2,2,3,3,4,9,2,2,2,2,
%T A196930 2,2,2,3,3,4,5,10,2,2,2,2,2,2,2,2,3,3,3,4,5,11,2,2,2,2,2,2,2,2,2,2,2,
%U A196930 2,3,3,3,3,4,4,5,6,12,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,6,13
%N A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.
%C A196930 For n >= 2, row n lists the parts of the head of the last section of the set of partitions of n, except the emergent parts.
%C A196930 Also 1 together with the integers > 1 of A196931.
%H A196930 Alois P. Heinz, <a href="/A196930/b196930.txt">Rows n = 1..33, flattened</a>
%e A196930 Written as a triangle:
%e A196930 1,
%e A196930 2,
%e A196930 3,
%e A196930 2,4,
%e A196930 2,5,
%e A196930 2,2,3,6
%e A196930 2,2,3,7,
%e A196930 2,2,2,2,3,4,8,
%e A196930 2,2,2,2,3,3,4,9,
%e A196930 2,2,2,2,2,2,2,3,3,4,5,10,
%e A196930 2,2,2,2,2,2,2,2,3,3,3,4,5,11,
%e A196930 2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6,12,
%e A196930 2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,6,13,
%e A196930 ...
%e A196930 Row n has length A002865(n), n >= 2. The sum of row n is A182708(n), n >= 2. The number of 2's in row n is A002865(n-2), n >= 4. Right border of triangle gives A000027.
%p A196930 p:= (f, g)-> zip((x, y)->x+y, f, g, 0):
%p A196930 b:= proc(n, i) option remember; local g, j, r;
%p A196930       if n=0 then [1] elif i<2 then [0]
%p A196930     else r:= b(n, i-1);
%p A196930          for j to n/i do g:= b(n-i*j, i-1);
%p A196930            r:= p(p(r, [0$i, g[1]]), subsop(1=0, g));
%p A196930          od; r
%p A196930       fi
%p A196930     end:
%p A196930 T:= proc(n) local l; l:= b(n$2);
%p A196930       `if`(n=1, 1, seq(i$l[i+1], i=2..nops(l)-1))
%p A196930     end:
%p A196930 seq(T(n), n=1..16);  # _Alois P. Heinz_, May 30 2013
%t A196930 p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{ g, j, r}, Which[n == 0, {1}, i<2, {0}, True, r = b[n, i-1]; For[j = 1, j <= n/i, j++, g = b[n-i*j, i-1]; r = p[p[r, Append[Array[0&, i], g // First]], ReplacePart[g, 1 -> 0]]]; r]]; T[n_] := Module[{l}, l = b[n, n]; If[n == 1, {1}, Table[Array[i&, l[[i+1]]], {i, 2, Length[l]-1}] // Flatten]]; Table[T[n], {n, 1, 16}] // Flatten (* _Jean-François Alcover_, Jan 30 2014, after _Alois P. Heinz_ *)
%Y A196930 Where records occur give A000041.
%Y A196930 Cf. A002865, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196931.
%K A196930 nonn,tabf
%O A196930 1,2
%A A196930 _Omar E. Pol_, Oct 21 2011