%I #24 Jan 30 2014 09:21:42

%S 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 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 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 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 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 Also 1 together with the integers > 1 of A196931.

%H Alois P. Heinz, <a href="/A196930/b196930.txt">Rows n = 1..33, flattened</a>

%e Written as a triangle:

%e 1,

%e 2,

%e 3,

%e 2,4,

%e 2,5,

%e 2,2,3,6

%e 2,2,3,7,

%e 2,2,2,2,3,4,8,

%e 2,2,2,2,3,3,4,9,

%e 2,2,2,2,2,2,2,3,3,4,5,10,

%e 2,2,2,2,2,2,2,2,3,3,3,4,5,11,

%e 2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6,12,

%e 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 ...

%e 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 p:= (f, g)-> zip((x, y)->x+y, f, g, 0):

%p b:= proc(n, i) option remember; local g, j, r;

%p if n=0 then [1] elif i<2 then [0]

%p else r:= b(n, i-1);

%p for j to n/i do g:= b(n-i*j, i-1);

%p r:= p(p(r, [0$i, g[1]]), subsop(1=0, g));

%p od; r

%p fi

%p end:

%p T:= proc(n) local l; l:= b(n$2);

%p `if`(n=1, 1, seq(i$l[i+1], i=2..nops(l)-1))

%p end:

%p seq(T(n), n=1..16); # _Alois P. Heinz_, May 30 2013

%t 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 Where records occur give A000041.

%Y Cf. A002865, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196931.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Oct 21 2011