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A196930
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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.
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5
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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