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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. 5
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. Also 1 together with the integers > 1 of A196931. LINKS Alois P. Heinz, Rows n = 1..33, flattened EXAMPLE Written as a triangle: 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, 2,2,2,2,2,2,2,3,3,4,5,10, 2,2,2,2,2,2,2,2,3,3,3,4,5,11, 2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6,12, 2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,6,13, ... 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. MAPLE p:= (f, g)-> zip((x, y)->x+y, f, g, 0): b:= proc(n, i) option remember; local g, j, r; if n=0 then [1] elif i<2 then [0] else r:= b(n, i-1); for j to n/i do g:= b(n-i*j, i-1); r:= p(p(r, [0\$i, g[1]]), subsop(1=0, g)); od; r fi end: T:= proc(n) local l; l:= b(n\$2); `if`(n=1, 1, seq(i\$l[i+1], i=2..nops(l)-1)) end: seq(T(n), n=1..16); # Alois P. Heinz, May 30 2013 MATHEMATICA 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 *) CROSSREFS Where records occur give A000041. Cf. A002865, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196931. Sequence in context: A060937 A162322 A336155 * A305437 A111982 A323236 Adjacent sequences: A196927 A196928 A196929 * A196931 A196932 A196933 KEYWORD nonn,tabf AUTHOR Omar E. Pol, Oct 21 2011 STATUS approved

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Last modified June 5 18:54 EDT 2023. Contains 363138 sequences. (Running on oeis4.)