Search: a363621 -id:a363621
|
|
A363532
|
|
Number of integer partitions of n with weighted alternating sum 0.
|
|
+10
9
|
|
|
1, 0, 0, 1, 0, 0, 2, 2, 0, 3, 3, 3, 5, 5, 10, 12, 7, 14, 25, 18, 22, 48, 48, 41, 67, 82, 89, 111, 140, 170, 220, 214, 264, 392, 386, 436, 623, 693, 756, 934, 1102, 1301, 1565, 1697, 2132, 2616, 2727, 3192, 4062, 4550, 5000, 6132, 7197, 8067, 9338, 10750, 12683
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,7
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
The a(11) = 3 through a(15) = 12 partitions (A = 10):
(33221) (84) (751) (662) (A5)
(44111) (6222) (5332) (4442) (8322)
(222221) (7311) (6421) (5531) (9411)
(621111) (532111) (43331) (722211)
(51111111) (42211111) (54221) (831111)
(65111) (3322221)
(432221) (3333111)
(443111) (4422111)
(32222111) (5511111)
(33311111) (22222221)
(72111111)
(6111111111)
|
|
MATHEMATICA
|
altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]], {k, 1, Length[y]}];
Table[Length[Select[IntegerPartitions[n], altwtsum[#]==0&]], {n, 0, 30}]
|
|
CROSSREFS
|
These partitions have ranks A363621.
The version for compositions is A363626.
A363619 gives weighted alternating sum of prime indices, reverse A363620.
A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A363626
|
|
Number of integer compositions of n with weighted alternating sum 0.
|
|
+10
9
|
|
|
1, 0, 0, 1, 1, 0, 2, 5, 7, 8, 14, 38, 64, 87, 174, 373, 649, 1069, 2051, 4091, 7453, 13276, 25260, 48990, 91378, 168890, 321661, 618323, 1169126, 2203649, 4211163, 8085240, 15421171, 29390131, 56382040, 108443047, 208077560, 399310778
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,7
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
The a(3) = 1 through a(10) = 14 compositions:
(21) (121) . (42) (331) (242) (63) (541)
(3111) (1132) (1331) (153) (2143)
(2221) (11132) (4122) (3232)
(21121) (12221) (5211) (4321)
(112111) (23111) (13122) (15112)
(121121) (14211) (31231)
(1112111) (411111) (42121)
(1311111) (114112)
(212122)
(213211)
(311221)
(322111)
(3111121)
(21211111)
|
|
MATHEMATICA
|
altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]], {k, 1, Length[y]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], altwtsum[#]==0&]], {n, 0, 10}]
|
|
CROSSREFS
|
A363619 gives weighted alternating sum of prime indices, reverse A363620.
A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|
|
A363619
|
|
Weighted alternating sum of the multiset of prime indices of n.
|
|
+10
8
|
|
|
0, 1, 2, -1, 3, -3, 4, 2, -2, -5, 5, 5, 6, -7, -4, -2, 7, 3, 8, 8, -6, -9, 9, -6, -3, -11, 4, 11, 10, 6, 11, 3, -8, -13, -5, -3, 12, -15, -10, -10, 13, 9, 14, 14, 7, -17, 15, 8, -4, 4, -12, 17, 16, -5, -7, -14, -14, -19, 17, -7, 18, -21, 10, -3, -9, 12, 19, 20
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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.
|
|
LINKS
|
|
|
EXAMPLE
|
The prime indices of 300 are {1,1,2,3,3}, with weighted alternating sum 1*1 - 2*1 + 3*2 - 4*3 + 5*3 = 8, so a(300) = 8.
|
|
MATHEMATICA
|
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]], {k, 1, Length[y]}];
Table[altwtsum[prix[n]], {n, 100}]
|
|
CROSSREFS
|
A359677 gives zero-based weighted sum of prime indices, reverse A359674.
Cf. A000009, A000720, A001221, A046660, A053632, A106529, A124010, A181819, A261079, A363532, A363621, A363626.
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A363620
|
|
Reverse-weighted alternating sum of the multiset of prime indices of n.
|
|
+10
8
|
|
|
0, 1, 2, 1, 3, 0, 4, 2, 2, -1, 5, 3, 6, -2, 1, 2, 7, 1, 8, 4, 0, -3, 9, 1, 3, -4, 4, 5, 10, 2, 11, 3, -1, -5, 2, 3, 12, -6, -2, 0, 13, 3, 14, 6, 5, -7, 15, 4, 4, 0, -3, 7, 16, 0, 1, -1, -4, -8, 17, 2, 18, -9, 6, 3, 0, 4, 19, 8, -5, 1, 20, 2, 21, -10, 3, 9, 3
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
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}.
|
|
LINKS
|
|
|
EXAMPLE
|
The prime indices of 300 are {1,1,2,3,3}, with reverse-weighted alternating sum 1*3 - 2*3 + 3*2 - 4*1 + 5*1 = 4, so a(300) = 4.
|
|
MATHEMATICA
|
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]], {k, 1, Length[y]}];
Table[revaltwtsum[prix[n]], {n, 100}]
|
|
CROSSREFS
|
Cf. A001221, A046660, A053632, A106529, A124010, A222855, A261079, A358137, A359674, A359755, A363531, A363532.
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A363622
|
|
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).
|
|
+10
8
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,11
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
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)
|
|
MATHEMATICA
|
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]]}]
|
|
CROSSREFS
|
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.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A363623
|
|
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).
|
|
+10
8
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,6
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
EXAMPLE
|
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)
|
|
MATHEMATICA
|
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]}]
|
|
CROSSREFS
|
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.
|
|
KEYWORD
|
nonn,tabf
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A363624
|
|
Weighted alternating sum of the integer partition with Heinz number n.
|
|
+10
8
|
|
|
0, 1, 2, -1, 3, 0, 4, 2, -2, 1, 5, 3, 6, 2, -1, -2, 7, 1, 8, 4, 0, 3, 9, -1, -3, 4, 4, 5, 10, 2, 11, 3, 1, 5, -2, -3, 12, 6, 2, 0, 13, 3, 14, 6, 5, 7, 15, 4, -4, 0, 3, 7, 16, 0, -1, 1, 4, 8, 17, -2, 18, 9, 6, -3, 0, 4, 19, 8, 5, 1, 20, 2, 21, 10, 3, 9, -3, 5
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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.
|
|
LINKS
|
|
|
EXAMPLE
|
The partition with Heinz number 600 is (3,3,2,1,1,1), with weighted alternating sum 1*3 - 2*3 + 3*2 - 4*1 + 5*1 - 6*1 = -2, so a(600) = -2.
|
|
MATHEMATICA
|
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
altwtsum[y_]:=Sum[(-1)^(k-1)*k*y[[k]], {k, 1, Length[y]}];
Table[altwtsum[Reverse[prix[n]]], {n, 100}]
|
|
CROSSREFS
|
For multisets instead of partitions we have A363619.
A359677 gives zero-based weighted sum of prime indices, reverse A359674.
A363626 counts compositions with reverse-weighted alternating sum 0.
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A363625
|
|
Reverse-weighted alternating sum of the integer partition with Heinz number n.
|
|
+10
8
|
|
|
0, 1, 2, 1, 3, 3, 4, 2, 2, 5, 5, 5, 6, 7, 4, 2, 7, 3, 8, 8, 6, 9, 9, 6, 3, 11, 4, 11, 10, 6, 11, 3, 8, 13, 5, 3, 12, 15, 10, 10, 13, 9, 14, 14, 7, 17, 15, 8, 4, 4, 12, 17, 16, 5, 7, 14, 14, 19, 17, 7, 18, 21, 10, 3, 9, 12, 19, 20, 16, 7, 20, 4, 21, 23, 5, 23
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
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}.
|
|
LINKS
|
|
|
EXAMPLE
|
The partition with Heinz number 600 is (3,3,2,1,1,1), so a(600) = -1*1 + 2*1 - 3*1 + 4*2 - 5*3 + 6*3 = 9.
|
|
MATHEMATICA
|
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]], {k, 1, Length[y]}];
Table[revaltwtsum[Reverse[prix[n]]], {n, 100}]
|
|
CROSSREFS
|
For multisets instead of partitions we have A363620.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
Search completed in 0.006 seconds
|