A034691 -id:A034691 - cp's OEIS frontend

A299151 Numerators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

1, 1, 2, 7, 8, 14, 32, 121, 126, 248, 512, 1003, 2048, 4064, 8176, 130539, 32768, 65382, 131072, 261868, 524224, 1048064, 2097152, 4193131, 8388576, 16775168, 33554180, 67104688, 134217728, 268426672, 536870912, 8589802359, 2147482624, 4294934528, 8589934336, 17179801257, 34359738368, 68719345664, 137438949376, 274877643724, 549755813888

Offset: 1

Gus Wiseman, Feb 03 2018

Numerators of rational valued sequence f whose Dirichlet convolution with itself yields function g(n) = A000079(n-1) = 2^(n-1). - Antti Karttunen, Aug 10 2018

			Sequence begins: 1, 1, 2, 7/2, 8, 14, 32, 121/2, 126, 248, 512, 1003, 2048, 4064, 8176, 130539/8, 32768.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000740, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299119, A299149, A299152.

Cf. also A317831.

nn=50;
sys=Table[2^(n-1)==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Numerator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]

A299151perA299152(n) = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299151perA299152(d)*A299151perA299152(n/d),0)))/2);
A299151(n) = numerator(A299151perA299152(n));

More terms from Antti Karttunen, Jul 29 2018

A306007 Number of non-isomorphic intersecting antichains of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 6, 6, 14, 22

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting antichain S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection, and none of which is a subset of any other. The weight of S is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(8) = 14 set-systems:
{{1,2,3,4,5,6,7,8}}
{{1,7},{2,3,4,5,6,7}}
{{1,2,7},{3,4,5,6,7}}
{{1,5,6},{2,3,4,5,6}}
{{1,2,3,7},{4,5,6,7}}
{{1,2,5,6},{3,4,5,6}}
{{1,3,4,5},{2,3,4,5}}
{{1,2},{1,3,4},{2,3,4}}
{{1,4},{1,5},{2,3,4,5}}
{{1,5},{2,4,5},{3,4,5}}
{{1,6},{2,6},{3,4,5,6}}
{{1,6},{2,3,6},{4,5,6}}
{{2,4},{1,2,5},{3,4,5}}
{{1,5},{2,5},{3,5},{4,5}}

Cf. A007411, A007716, A034691, A048143, A049311, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306008.

A335469 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,1,2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Jun 16 2020

nonn

First differs from A374701 in having 150, corresponding to (3,2,1,2). - Gus Wiseman, Sep 18 2024
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			See A335468 for an example of the complement.

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A335468 is the matching version.
The (1,2,1)-avoiding version is A335467.
These compositions are counted by A335473.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134 and ranked by A334030.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A001710, A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335450, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,y_,_,x_,_}/;x>y]&]

A334269 Number of compositions of n that are both a reversed Lyndon word and a co-Lyndon word.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 16, 23, 40, 62, 110, 169, 302, 492, 856, 1454, 2572, 4428, 7914, 13935, 25036, 44842, 81298, 147149, 268952, 491746, 904594, 1667091, 3085950, 5723367, 10652544, 19865887, 37150314, 69608939, 130723184, 245935633, 463590444, 875306913, 1655451592, 3135613649, 5948011978, 11298215516

Offset: 1

Gus Wiseman, Apr 24 2020

nonn

Also the number of compositions of n that are both a Lyndon word and a reversed co-Lyndon word.
A composition of n is a finite sequence of positive integers summing to n.
A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations. Co-Lyndon is defined similarly, except with strictly greater instead of strictly less.

			The a(1) = 1 through a(7) = 16 compositions:
  (1)  (2)  (3)   (4)    (5)     (6)      (7)
            (21)  (31)   (32)    (42)     (43)
                  (211)  (41)    (51)     (52)
                         (221)   (321)    (61)
                         (311)   (411)    (322)
                         (2111)  (2211)   (331)
                                 (3111)   (421)
                                 (21111)  (511)
                                          (2221)
                                          (3121)
                                          (3211)
                                          (4111)
                                          (21211)
                                          (22111)
                                          (31111)
                                          (211111)

The version for binary expansion is A334267.
Compositions of this type are ranked by A334266.
Normal sequences of this type are counted by A334270.
Necklace compositions of this type are counted by A334271.
Aperiodic compositions are counted by A000740.
Binary Lyndon words are counted by A001037.
Necklace compositions are counted by A008965.
Normal Lyndon words are counted by A060223.
Lyndon compositions are counted by A059966.
All of the following pertain to compositions in standard order (A066099):
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed Lyndon words are A334265.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Length of co-Lyndon factorization is A334029.
- Length of Lyndon factorization of reverse is A334297.
- Length of co-Lyndon factorization of reverse is A329313.
- Lyndon factorizations are counted by A333940.
- Co-Lyndon factorizations are counted by A333765.
- Aperiodic compositions are A328594.
- Distinct rotations are counted by A333632.
Cf. A034691, A065609, A275692, A328596, A329141, A329324, A329326, A334266, A334272, A334273, A334274.

lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],lynQ[Reverse[#]]&&colynQ[#]&]],{n,0,15}]

Offset corrected and a(21)-a(42) from Bert Dobbelaere, Apr 26 2020

A334271 Number of compositions of n that are both a reversed necklace and a co-necklace.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 17, 28, 43, 70, 111, 184, 303, 510, 865, 1482, 2573, 4480, 7915, 14008

Offset: 0

Gus Wiseman, Apr 25 2020

Also the number of compositions of n that are both a necklace and a reversed co-necklace.

A necklace is a finite sequence of positive integers that is lexicographically less than or equal to any cyclic rotation. Co-necklace is defined similarly, except with greater instead of less.

			The a(1) = 1 through a(6) = 12 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (21)   (22)    (32)     (33)
             (111)  (31)    (41)     (42)
                    (211)   (221)    (51)
                    (1111)  (311)    (222)
                            (2111)   (321)
                            (11111)  (411)
                                     (2121)
                                     (2211)
                                     (3111)
                                     (21111)
                                     (111111)

Normal sequences of this type are counted by A334272.
The aperiodic case is A334269.
These compositions are ranked by A334273.
Binary (or reversed binary) necklaces are counted by A000031.
Normal sequences are counted by A000670.
Necklace compositions are counted by A008965.
Lyndon compositions are counted by A059966.
Normal Lyndon words are counted by A060223.
Normal necklaces are counted by A019536.
Normal aperiodic words are counted by A296975.
All of the following pertain to compositions in standard order (A066099):
- Necklaces are A065609.
- Reversed necklaces are A333943.
- Co-necklaces are A333764.
- Reversed co-necklaces are A328595.
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed Lyndon words are A334265.
- Reversed co-Lyndon words are A328596.
- Aperiodic compositions are A328594.
Cf. A000740, A001037, A034691, A329138, A329324, A334266, A334267, A334274.

neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[Reverse[#]]&&coneckQ[#]&]],{n,0,15}]

A334272 Number of sequences of length n that cover an initial interval of positive integers and are both a reversed necklace and a co-necklace.

Original entry on oeis.org

1, 1, 2, 4, 12, 43, 229, 1506, 12392, 120443

Offset: 0

Gus Wiseman, Apr 25 2020

A necklace is a finite sequence of positive integers that is lexicographically strictly less than or equal to any cyclic rotation. Co-necklace is defined similarly, except with strictly greater instead of strictly less.

			The a(1) = 1 through a(4) = 12 normal sequences:
  (1)  (1,1)  (1,1,1)  (1,1,1,1)
       (2,1)  (2,1,1)  (2,1,1,1)
              (2,2,1)  (2,1,2,1)
              (3,2,1)  (2,2,1,1)
                       (2,2,2,1)
                       (3,1,2,1)
                       (3,2,1,1)
                       (3,2,2,1)
                       (3,2,3,1)
                       (3,3,2,1)
                       (4,2,3,1)
                       (4,3,2,1)

Dominates A334270 (the aperiodic case).
Compositions of this type are counted by A334271.
These compositions are ranked by A334273 (standard) and A334274 (binary).
Binary (or reversed binary) necklaces are counted by A000031.
Normal sequences are counted by A000670.
Necklace compositions are counted by A008965.
Normal Lyndon words are counted by A060223.
Normal necklaces are counted by A019536.
All of the following pertain to compositions in standard order (A066099):
- Necklaces are A065609.
- Reversed necklaces are A333943.
- Co-necklaces are A333764.
- Reversed co-necklaces are A328595.
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed Lyndon words are A334265.
- Reversed co-Lyndon words are A328596.
- Reversed Lyndon co-Lyndon compositions are A334266.
- Aperiodic compositions are A328594.
Cf. A034691, A059966, A296372, A296975, A329138, A334269.

neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],neckQ[Reverse[#]]&&coneckQ[#]&]],{n,0,8}]

A335483 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,1,2).

Original entry on oeis.org

38, 70, 77, 78, 102, 134, 140, 141, 142, 150, 154, 155, 157, 158, 166, 198, 205, 206, 230, 262, 268, 269, 270, 276, 278, 281, 282, 283, 284, 285, 286, 294, 301, 302, 306, 308, 309, 310, 311, 314, 315, 317, 318, 326, 333, 334, 358, 390, 396, 397, 398, 406, 410

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with the corresponding compositions begins:
   38: (3,1,2)
   70: (4,1,2)
   77: (3,1,2,1)
   78: (3,1,1,2)
  102: (1,3,1,2)
  134: (5,1,2)
  140: (4,1,3)
  141: (4,1,2,1)
  142: (4,1,1,2)
  150: (3,2,1,2)
  154: (3,1,2,2)
  155: (3,1,2,1,1)
  157: (3,1,1,2,1)
  158: (3,1,1,1,2)
  166: (2,3,1,2)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Permutations matching (1,3,2,4) are counted by A158009.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,z_,_}/;y

A335486 Numbers k such that the k-th composition in standard order (A066099) is not weakly increasing.

Original entry on oeis.org

5, 9, 11, 13, 17, 18, 19, 21, 22, 23, 25, 27, 29, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

Also compositions matching the pattern (2,1).

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence of terms together with the corresponding compositions begins:
   5: (2,1)
   9: (3,1)
  11: (2,1,1)
  13: (1,2,1)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  33: (5,1)
  34: (4,2)
  35: (4,1,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A225620 is the avoiding version.
The (1,2)-matching version is A335485.
Patterns matching this pattern are counted by A002051 (by length).
Permutations of prime indices matching this pattern are counted by A008480(n) - 1.
These compositions are counted by A056823 (by sum).
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A334968, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_}/;x>y]&]

A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).

Original entry on oeis.org

1, 1, 5, 21, 95, 415, 1851, 8155, 36030, 158510, 696502, 3052966, 13359230, 58346206, 254405630, 1107479694, 4813850699, 20894227355, 90567536543, 392066476815, 1695180397145, 7320927664713, 31581573600685, 136094434672509, 585876330191950, 2519701493092958

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144068, A343350, A343351, A343352, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*4^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 12 2021

nmax = 25; CoefficientList[Series[Product[1/(1 - x^k)^(4^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 4^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

a(n) ~ exp(sqrt(n) - 1/8 + c/4) * 2^(2*n - 3/2) / (sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (4^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

A343350 Expansion of Product_{k>=1} 1 / (1 - x^k)^(5^(k-1)).

Original entry on oeis.org

1, 1, 6, 31, 171, 921, 5031, 27281, 148101, 801901, 4336902, 23415777, 126254962, 679805112, 3655679442, 19634501447, 105334380517, 564471596667, 3021754455157, 16160029793032, 86339725851558, 460874548444683, 2457961986888773, 13097958657023523, 69740119667456018

Offset: 0

Ilya Gutkovskiy, Apr 12 2021

nonn

Cf. A034691, A104460, A144069, A343349, A343351, A343352, A343353, A343354, A343355.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      d*5^(d-1), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Apr 12 2021

nmax = 24; CoefficientList[Series[Product[1/(1 - x^k)^(5^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, (1/n) Sum[Sum[d 5^(d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]]; Table[a[n], {n, 0, 24}]

a(n) ~ exp(2*sqrt(n/5) - 1/10 + c/5) * 5^(n - 1/4) / (2*sqrt(Pi)*n^(3/4)), where c = Sum_{j>=2} 1/(j * (5^(j-1) - 1)). - Vaclav Kotesovec, Apr 12 2021

cp's OEIS Frontend

A299151 Numerators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

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A306007 Number of non-isomorphic intersecting antichains of weight n.

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A335469 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,1,2).

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A334269 Number of compositions of n that are both a reversed Lyndon word and a co-Lyndon word.

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A334271 Number of compositions of n that are both a reversed necklace and a co-necklace.

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A334272 Number of sequences of length n that cover an initial interval of positive integers and are both a reversed necklace and a co-necklace.

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A335483 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (3,1,2).

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A335486 Numbers k such that the k-th composition in standard order (A066099) is not weakly increasing.

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A343349 Expansion of Product_{k>=1} 1 / (1 - x^k)^(4^(k-1)).

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