A183152
Irregular triangle read by rows in which row n lists the emergent parts of all partitions of n, or 0 if such parts do not exist.
Original entry on oeis.org
0, 0, 0, 0, 2, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 4, 7, 3, 6, 5, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 9, 4, 3, 3, 2, 2, 2, 2, 5, 4, 8, 4, 3, 7, 6
Offset: 0
If written as a triangle:
0,
0,
0,
0,
2,
3,
2,4,2,3,
3,5,2,4,
2,4,2,3,6,3,2,2,5,4,
3,5,2,4,7,3,2,2,3,6,3,5,
2,4,2,3,6,3,2,2,5,4,8,4,3,2,2,2,2,4,7,3,6,5,
3,5,2,4,7,3,2,2,3,6,3,5,9,4,3,3,2,2,2,2,5,4,8,4,3,7,6
A196025
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.
Original entry on oeis.org
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
Offset: 1
Cf.
A026905,
A046746,
A066186,
A135010,
A138121,
A182699,
A182707,
A182709,
A183152,
A193827,
A196039,
A196930,
A196931,
A198381.
A198381
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.
Original entry on oeis.org
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
Offset: 0
Cf.
A000041,
A000065,
A000070,
A006128,
A026905,
A093694,
A096541,
A135010,
A138121,
A182699,
A182707,
A182709,
A183152,
A193827,
A196930,
A196931.
A182708
a(n) is the sum of the smallest parts of all partitions of n that do not contain 1 as a part.
Original entry on oeis.org
0, 2, 3, 6, 7, 13, 14, 23, 27, 39, 45, 67, 75, 104, 125, 165, 194, 258, 302, 392, 467, 588, 700, 885, 1045, 1296, 1546, 1897, 2249, 2753, 3252, 3945, 4670, 5616, 6633, 7957, 9357, 11157, 13124, 15573, 18257, 21599, 25259, 29760, 34760, 40788, 47526, 55642, 64669, 75465, 87576, 101898, 117991, 136977, 158286
Offset: 1
-
Table[Total[{Min /@ IntegerPartitions[n, All, Range[2, n]]}, 2], {n, 55}] (* Robert Price, Aug 30 2020 *) (* Only suitable for n<100 *)
-
my(N=66, z='z+O('z^N)); gf=sum(k=1, N, k * z^k / prod(j=k, N, 1-z^j ) ) - z/eta(z); concat([0], Vec(gf)) \\ Joerg Arndt, Aug 31 2020
A196039
Total sum of the smallest part of every partition of every shell of n.
Original entry on oeis.org
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
Offset: 0
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.
Cf.
A026905,
A046746,
A066186,
A135010,
A138121,
A182699,
A182707,
A182709,
A183152,
A193827,
A196025,
A196930,
A196931,
A198381,
A206437.
-
b:= proc(n, i) option remember;
`if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(nAlois P. Heinz, Apr 03 2012
-
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 *)
Showing 1-5 of 5 results.
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