Search: a222970 -id:a222970
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A231429
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Number of partitions of 2n into distinct parts < n.
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+10
8
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1, 0, 0, 0, 0, 1, 2, 4, 8, 14, 22, 35, 53, 78, 113, 160, 222, 306, 416, 558, 743, 980, 1281, 1665, 2149, 2755, 3514, 4458, 5626, 7070, 8846, 11020, 13680, 16920, 20852, 25618, 31375, 38309, 46649, 56651, 68616, 82908, 99940, 120192, 144238, 172730, 206425
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OFFSET
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0,7
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COMMENTS
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Also the number of integer compositions of n with weighted sum 3*n, where the weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i * y_i. The a(0) = 1 through a(9) = 14 compositions are:
() . . . . (11111) (3111) (3211) (3311) (3411)
(11211) (11311) (4121) (4221)
(12121) (11411) (5112)
(21112) (12221) (11511)
(13112) (12321)
(21131) (13131)
(21212) (13212)
(111122) (21231)
(21312)
(22122)
(31113)
(111141)
(111222)
(112113)
(End)
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LINKS
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EXAMPLE
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a(5) = #{4+3+2+1} = 1;
a(6) = #{5+4+3, 5+4+2+1} = 2;
a(7) = #{6+5+3, 6+5+2+1, 6+4+3+1, 5+4+3+2} = 4;
a(8) = #{7+6+3, 7+6+2+1, 7+6+3, 7+5+3+1, 7+4+3+2, 6+5+4+1, 6+5+3+2, 6+4+3+2+1} = 8;
a(9) = #{8+7+3, 8+7+2+1, 8+6+4, 8+6+3+1, 8+5+4+1, 8+5+3+2, 8+4+3+2+1, 7+6+5, 7+6+4+1, 7+6+3+2, 7+5+4+2, 7+5+3+2+1, 6+5+4+3, 6+5+4+2+1} = 14.
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Total[Accumulate[#]]==3n&]], {n, 0, 15}] (* Gus Wiseman, Jun 17 2023 *)
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PROG
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(Haskell)
a231429 n = p [1..n-1] (2*n) where
p _ 0 = 1
p [] _ = 0
p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
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CROSSREFS
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A053632 counts compositions by weighted sum.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A363622
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Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with weighted alternating sum k (leading and trailing 0's omitted).
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+10
8
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1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 1, 3, 0, 0, 2, 0, 1, 1, 2, 1, 1, 3, 0, 2, 2, 1, 1, 2, 2, 1, 1, 5, 0, 0, 3, 0, 2, 2, 2, 1, 3, 2, 1, 1, 5, 0, 3, 3, 2, 2, 3, 2, 2, 4, 2, 1, 1, 7, 0, 0, 5, 0, 3, 3, 4, 2, 4, 2, 4, 4, 2, 1, 1
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OFFSET
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0,11
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COMMENTS
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We define the weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(i-1) i * y_i. For example:
- (3,3,2,1,1) has weighted alternating sum 1*3 - 2*3 + 3*2 - 4*1 + 5*1 = 4.
- (1,2,2,3) has weighted alternating sum 1*1 - 2*2 + 3*2 - 4*3 = -9.
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LINKS
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EXAMPLE
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Triangle begins:
1
1
1 0 0 1
1 0 1 1
2 0 0 1 0 1 1
2 0 1 1 1 1 1
3 0 0 2 0 1 1 2 1 1
3 0 2 2 1 1 2 2 1 1
5 0 0 3 0 2 2 2 1 3 2 1 1
5 0 3 3 2 2 3 2 2 4 2 1 1
7 0 0 5 0 3 3 4 2 4 2 4 4 2 1 1
7 0 5 5 3 3 5 4 3 5 3 5 4 2 1 1
Row n = 6 counts the following partitions:
k=-3 k=0 k=2 k=3 k=4 k=5 k=6
-----------------------------------------------------------
(33) . . (42) . (321) (51) (222) (411) (6)
(2211) (3111) (21111)
(111111)
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MATHEMATICA
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altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]], {k, 1, Length[y]}];
Table[Length[Select[IntegerPartitions[n], altwtsum[#]==k&]], {n, 0, 15}, {k, Min[altwtsum/@IntegerPartitions[n]], Max[altwtsum/@IntegerPartitions[n]]}]
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CROSSREFS
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The unweighted version is A103919 with leading zeros removed.
A053632 counts compositions by weighted sum.
A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
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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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A363623
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Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with reverse-weighted alternating sum k (leading and trailing 0's omitted).
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+10
8
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1, 1, 1, 1, 1, 2, 2, 0, 1, 2, 2, 1, 1, 1, 1, 1, 3, 1, 0, 3, 0, 1, 1, 1, 1, 3, 2, 0, 3, 1, 2, 0, 1, 0, 1, 2, 5, 1, 0, 3, 1, 2, 2, 2, 1, 1, 0, 1, 0, 1, 2, 5, 3, 0, 4, 2, 2, 0, 3, 2, 1, 3, 0, 0, 1, 0, 1, 1, 1, 1, 7, 2, 0, 4, 1, 5, 2, 3, 1, 3, 0, 2, 3, 1, 2, 1, 0, 0, 1, 0, 1, 1, 1, 1
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OFFSET
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0,6
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COMMENTS
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We define the reverse-weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(k-i) i * y_{k-i+1}. For example:
- (3,3,2,1,1) has reverse-weighted alternating sum 1*1 - 2*1 + 3*2 - 4*3 + 5*3 = 8.
- (1,2,2,3) has reverse-weighted alternating sum -1*3 + 2*2 - 3*2 + 4*1 = -1.
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LINKS
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EXAMPLE
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Triangle begins:
1
1
1 1
1 2
2 0 1 2
2 1 1 1 1 1
3 1 0 3 0 1 1 1 1
3 2 0 3 1 2 0 1 0 1 2
5 1 0 3 1 2 2 2 1 1 0 1 0 1 2
5 3 0 4 2 2 0 3 2 1 3 0 0 1 0 1 1 1 1
Row n = 6 counts the following partitions:
k=3 k=4 k=6 k=8 k=9 k=10 k=11
--------------------------------------------------------------
(33) (222) . (6) . (21111) (51) (3111) (411)
(2211) (42)
(111111) (321)
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MATHEMATICA
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revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]], {k, 1, Length[y]}];
Table[Length[Select[IntegerPartitions[n], revaltwtsum[#]==k&]], {n, 0, 15}, {k, Floor[(n+1)/2], Ceiling[n*(n+1)/4]}]
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CROSSREFS
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Column k = floor((n+1)/2) is A119620.
A053632 counts compositions by weighted sum.
A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
Cf. A008284, A067538, A222855, A222970, A318283, A320387, A360672, A360675, A362559, A363532, A363621, A363626.
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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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A363526
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Number of integer partitions of n with reverse-weighted sum 3*n.
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+10
6
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1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 3, 2, 4, 4, 4, 5, 5, 4, 7, 7, 5, 8, 7, 6, 11, 9, 8, 11, 10, 10, 13, 12, 11, 15, 15, 12, 17, 16, 14, 20, 18, 16, 22, 20, 19, 24, 22, 20, 27, 26, 23, 29, 27, 25, 33, 30, 28, 35, 33, 31, 38, 36, 33, 41, 40
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OFFSET
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0,11
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COMMENTS
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Are the partitions counted all of length 4 or 5?
The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.
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LINKS
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EXAMPLE
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The partition (6,4,4,1) has sum 15 and reverse-weighted sum 45 so is counted under a(15).
The a(n) partitions for n = {5, 10, 15, 16, 21, 24}:
(1,1,1,1,1) (4,3,2,1) (6,4,4,1) (6,5,4,1) (8,6,6,1) (9,7,7,1)
(2,2,2,2,2) (6,5,2,2) (6,6,2,2) (8,7,4,2) (9,8,5,2)
(7,3,3,2) (7,4,3,2) (9,5,5,2) (9,9,3,3)
(3,3,3,3,3) (9,6,3,3) (10,6,6,2)
(10,4,4,3) (10,7,4,3)
(11,5,5,3)
(12,4,4,4)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Total[Accumulate[#]]==3n&]], {n, 0, 30}]
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CROSSREFS
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Positions of terms with omega > 4 appear to be A079998.
The version for compositions is A231429.
The non-reverse version is A363527.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
Cf. A000016, A008284, A067538, A222855, A222970, A359755, A360672, A360675, A362559, A362560, A363525, A363528.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A363527
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Number of integer partitions of n with weighted sum 3*n.
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+10
5
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1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 3, 4, 4, 6, 8, 7, 10, 13, 13, 21, 25, 24, 37, 39, 40, 58, 63, 72, 94, 106, 118, 144, 165, 181, 224, 256, 277, 341, 387, 417, 504, 560, 615, 743, 818, 899, 1066, 1171, 1285, 1502, 1655, 1819, 2108, 2315, 2547, 2915
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OFFSET
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0,15
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COMMENTS
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Are the partitions counted all of length > 4?
The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.
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LINKS
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EXAMPLE
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The partition (2,2,1,1,1,1) has sum 8 and weighted sum 24 so is counted under a(8).
The a(13) = 1 through a(18) = 8 partitions:
(332221) (333221) (33333) (442222) (443222) (443331)
(4322111) (522222) (5322211) (4433111) (444222)
(71111111) (4332111) (55111111) (5332211) (533322)
(63111111) (63211111) (55211111) (4443111)
(63311111) (7222221)
(72221111) (55311111)
(64221111)
(A11111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Total[Accumulate[Reverse[#]]]==3n&]], {n, 0, 30}]
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CROSSREFS
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The version for compositions is A231429.
These partitions have ranks A363531.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
Cf. A000016, A008284, A067538, A222855, A222970, A359755, A360672, A360675, A362559, A362560, A363525, A363528, A363532.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A363525
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Number of integer partitions of n with weighted sum divisible by reverse-weighted sum.
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+10
3
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1, 2, 2, 3, 2, 4, 2, 4, 5, 5, 3, 10, 4, 7, 13, 10, 8, 29, 10, 18, 39, 20, 20, 70, 29, 40, 105, 65, 55, 166, 73, 132, 242, 141, 129, 476, 183, 248, 580, 487, 312, 984, 422, 868, 1345, 825, 724, 2709, 949, 1505, 2756, 2902, 1611, 4664, 2289, 4942, 5828, 4278
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OFFSET
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1,2
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COMMENTS
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The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
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LINKS
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EXAMPLE
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The partition (6,5,4,3,2,1,1,1,1) has weighted sum 80, reverse 160, so is counted under a(24).
The a(n) partitions for n = 1, 2, 4, 6, 9, 12, 14 (A..E = 10-14):
1 2 4 6 9 C E
11 22 33 333 66 77
1111 222 711 444 65111
111111 6111 921 73211
111111111 3333 2222222
7311 71111111
63111 11111111111111
222222
621111
111111111111
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Divisible[Total[Accumulate[#]], Total[Accumulate[Reverse[#]]]]&]], {n, 30}]
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CROSSREFS
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The case of equality (and reciprocal version) is A000005.
A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
Cf. A000016, A008284, A067538, A222855, A222970, A358137, A359755, A362558, A362559, A362560, A363527.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A363528
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Number of strict integer partitions of n with weighted sum divisible by reverse-weighted sum.
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+10
3
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 6, 2, 3, 9, 3, 4, 11, 4, 5, 16, 6, 8, 24, 8, 10, 31, 11, 14, 41, 18, 18, 59, 21, 27, 74, 30, 32, 100, 35, 43, 128, 54, 53, 173, 58, 78, 215, 81, 88, 294, 97, 123, 362, 150, 146, 469, 162, 221, 577
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OFFSET
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1,12
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COMMENTS
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The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is also the sum of partial sums of the reverse.
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LINKS
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EXAMPLE
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The a(n) partitions for n = 1, 12, 15, 21, 24, 26:
(1) (12) (15) (21) (24) (26)
(9,2,1) (11,3,1) (15,5,1) (17,6,1) (11,8,4,2,1)
(9,3,2,1) (16,3,2) (18,4,2) (12,6,5,2,1)
(11,7,2,1) (12,9,2,1) (13,5,4,3,1)
(12,5,3,1) (13,7,3,1)
(10,5,3,2,1) (14,5,4,1)
(15,4,3,2)
(10,8,3,2,1)
(11,6,4,2,1)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Divisible[Total[Accumulate[#]], Total[Accumulate[Reverse[#]]]]&]], {n, 30}]
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CROSSREFS
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A318283 gives weighted sum of reversed prime indices, row-sums of A358136.
Cf. A008284, A053632, A067538, A222855, A222970, A358137, A359754, A359755, A362558, A362559, A362560.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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