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Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n.
+0
9
0, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
COMMENTS
If n >= 1, row n lists the smallest parts of every partition of n in the order produced by the shell model of partitions of A135010, hence row n lists the parts of the last section of the set of partitions of n, except the emergent parts (See A182699).
EXAMPLE
Written as a triangle:
0,
1,
1,2,
1,1,3,
1,1,1,2,4,
1,1,1,1,1,2,5,
1,1,1,1,1,1,1,2,2,3,6
1,1,1,1,1,1,1,1,1,1,1,2,2,3,7,
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,8,
...
Total sum of parts greater than 1 in all the partitions of n except one copy of the smallest part greater than 1 of every partition.
+0
5
0, 0, 0, 2, 5, 16, 30, 63, 108, 189, 298, 483, 720, 1092, 1582, 2297, 3225, 4551, 6244, 8592, 11590, 15622, 20741, 27536, 36066, 47198, 61150, 79077, 101391, 129808, 164934, 209213, 263745, 331807, 415229, 518656, 644719, 799926, 988432, 1218979
COMMENTS
Also partial sums of A182709. Total sum of emergent parts in all partitions of all numbers <= n. Also total sum of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012
CROSSREFS
Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196039, A196930, A196931, A198381.
Total sum of the smallest part of every partition of every shell of n.
+0
2
0, 1, 4, 9, 18, 30, 50, 75, 113, 162, 231, 318, 441, 593, 798, 1058, 1399, 1824, 2379, 3066, 3948, 5042, 6422, 8124, 10264, 12884, 16138, 20120, 25027, 30994, 38312, 47168, 57955, 70974, 86733, 105676, 128516, 155850, 188644, 227783, 274541
COMMENTS
Total sum of parts of all regions of n that contain 1 as a part. - Omar E. Pol, Mar 11 2012
EXAMPLE
For n = 5 the seven partitions of 5 are:
5
3 + 2
4 + 1
2 + 2 + 1
3 + 1 + 1
2 + 1 + 1 + 1
1 + 1 + 1 + 1 + 1
.
The five shells of 5 (see A135010 and also A138121), written as a triangle, are: 1
2, 1
3, 1, 1
4, (2, 2), 1, 1, 1
5, (3, 2), 1, 1, 1, 1, 1
.
The first "2" of row 4 does not count, also the "3" of row 5 does not count, so we have:
1
2, 1
3, 1, 1
4, 2, 1, 1, 1
5, 2, 1, 1, 1, 1, 1
.
thus a(5) = 1+2+1+3+1+1+4+2+1+1+1+5+2+1+1+1+1+1 = 30.
MAPLE
b:= proc(n, i) option remember;
`if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(n<i, 0, b(n-i, i)))
end:
a:= proc(n) option remember;
b(n, n) +`if`(n=0, 0, a(n-1))
end:
MATHEMATICA
b[n_, i_] := b[n, i] = If[n == i, n, 0] + If[i < 1, 0, b[n, i-1] + If[n < i, 0, b[n-i, i]]]; Accumulate[Table[b[n, n], {n, 0, 50}]] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
CROSSREFS
Cf. A026905, A046746, A066186, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196025, A196930, A196931, A198381, A206437.
Total number of parts greater than 1 in all partitions of n minus the number of partitions of n into parts each less than n.
+0
5
0, 0, 0, 0, 1, 2, 6, 10, 20, 32, 54, 81, 128, 184, 273, 385, 549, 754, 1048, 1412, 1917, 2547, 3392, 4444, 5837, 7556, 9791, 12553, 16086, 20429, 25935, 32665, 41108, 51404, 64190, 79721, 98882, 122043, 150417, 184618, 226239
COMMENTS
Also partial sums of A182699. Total number of emergent parts in all partitions of the numbers <= n. Also total number of parts of all regions of n that do not contain 1 as a part (Cf. A083751, A187219). - Omar E. Pol, Mar 04 2012
CROSSREFS
Cf. A000041, A000065, A000070, A006128, A026905, A093694, A096541, A135010, A138121, A182699, A182707, A182709, A183152, A193827, A196930, A196931.
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