A032021 -id:A032021 - cp's OEIS frontend

A331846 Number of compositions (ordered partitions) of n into distinct squarefree parts.

1, 1, 1, 3, 2, 3, 9, 5, 12, 16, 21, 41, 42, 49, 59, 79, 130, 231, 230, 295, 226, 495, 609, 699, 1472, 1042, 1377, 2308, 2982, 3425, 3879, 4877, 7156, 7189, 13531, 14797, 13570, 19551, 27667, 30327, 36382, 47519, 60783, 70561, 78330, 136988, 121659, 174851

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(7) = 5 because we have [7], [6, 1], [5, 2], [2, 5] and [1, 6].

Index entries for sequences related to compositions

Cf. A005117, A032020, A032021, A032022, A087188, A218396, A219107, A280194, A331843, A331844, A331845, A331847.

A331847 Number of compositions (ordered partitions) of n into distinct prime powers (1 excluded).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 2, 5, 3, 11, 10, 13, 18, 19, 52, 30, 61, 77, 114, 109, 146, 260, 318, 341, 356, 631, 666, 927, 848, 1849, 1978, 2305, 2213, 3560, 4302, 4748, 5588, 6779, 13952, 9044, 15534, 16897, 25084, 20731, 29524, 34882, 49360, 50765, 55112, 106903, 83652, 128552, 106638

Offset: 0

Ilya Gutkovskiy, Jan 29 2020

nonn

			a(10) = 10 because we have [8, 2], [7, 3], [5, 3, 2], [5, 2, 3], [3, 7], [3, 5, 2], [3, 2, 5], [2, 8], [2, 5, 3] and [2, 3, 5].

Index entries for sequences related to compositions

Cf. A032020, A032021, A032022, A035942, A054685, A218396, A219107, A246655, A280195, A331843, A331844, A331845, A331846.

A332032 G.f.: Sum_{k>=1} k! * x^(k^2) / (1 - x^(2*k)).

Original entry on oeis.org

1, 0, 1, 2, 1, 0, 1, 2, 7, 0, 1, 2, 1, 0, 7, 26, 1, 0, 1, 2, 7, 0, 1, 26, 121, 0, 7, 2, 1, 0, 1, 26, 7, 0, 121, 722, 1, 0, 7, 26, 1, 0, 1, 2, 127, 0, 1, 746, 5041, 0, 7, 2, 1, 0, 121, 26, 7, 0, 1, 722, 1, 0, 5047, 40346, 121, 0, 1, 2, 7, 0, 1, 746, 1, 0, 127

Offset: 1

Ilya Gutkovskiy, Feb 05 2020

nonn

Number of compositions (ordered partitions) of n into distinct odd parts, where every odd part between the largest and smallest appears.

			a(9) = 7 because we have [9], [5, 3, 1], [5, 1, 3], [3, 5, 1], [3, 1, 5], [1, 5, 3] and [1, 3, 5].
a(12) = 2 because we have [7, 5], and [5, 7]. - _Antti Karttunen_, Dec 15 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A016825 (positions of 0's), A032021, A034178, A038548, A107461, A220400.

Coincides with A332031 on odd numbers.

nmax = 75; CoefficientList[Series[Sum[k! x^(k^2)/(1 - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A332032(n) = sumdiv(n, d, if(!((n+d)%2) && !((n+(n/d))%2) && (d<=(n/d)), d!, 0)); \\ Antti Karttunen, Dec 15 2021

From Antti Karttunen, Dec 15 2021: (Start)
a(n) = Sum_{d|n, d <= n/d, and both d and n/d are of the same parity as n} d!.
a(2n-1) = A332031(2n-1) for all n >= 1.
(End)

A218694 Carlitz compositions of n into odd parts.

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 2, 3, 6, 9, 10, 13, 22, 32, 40, 56, 86, 122, 164, 229, 332, 474, 656, 914, 1310, 1867, 2604, 3648, 5184, 7346, 10318, 14506, 20516, 29022, 40880, 57548, 81260, 114810, 161864, 228092, 321892, 454444, 640954, 903715, 1274998, 1799320, 2538218, 3579714, 5049954, 7125359, 10051844

Offset: 0

Joerg Arndt, Nov 04 2012

nonn

Carlitz compositions are compositions where adjacent parts are distinct (see A003242).

			There are a(12) = 22 such compositions of 12:
[ 1]  1 3 1 3 1 3
[ 2]  1 3 1 7
[ 3]  1 3 5 3
[ 4]  1 3 7 1
[ 5]  1 5 1 5
[ 6]  1 7 1 3
[ 7]  1 7 3 1
[ 8]  1 11
[ 9]  3 1 3 1 3 1
[10]  3 1 3 5
[11]  3 1 5 3
[12]  3 1 7 1
[13]  3 5 1 3
[14]  3 5 3 1
[15]  3 9
[16]  5 1 5 1
[17]  5 3 1 3
[18]  5 7
[19]  7 1 3 1
[20]  7 5
[21]  9 3
[22]  11 1

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms 0..262 from Joerg Arndt)

Cf. A003242 (Carlitz compositions), A032021 (compositions into distinct odd parts), A032020 (compositions into distinct parts).

b:= proc(n, t) option remember; `if`(n=0, 1,
       add(`if`(j=t or irem(j, 2)=0, 0, b(n-j, j)), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Nov 08 2012

nn=20;CoefficientList[Series[1/(1-Sum[z^(2j+1)/(1+z^(2j+1)),{j,0,nn}]),{z,0,nn}],z] (* Geoffrey Critzer, Nov 21 2013 *)

G.f.: 1/( 1 - Sum_{j>=0} x^(2j+1)/(1 + x^(2j+1)) ). - Geoffrey Critzer, Nov 21 2013

a(n) ~ c / r^n, where r = 0.708865489663179258570259601255070249415... is the root of the equation sum_{j>=0} x^(2j+1)/(1 + x^(2j+1)) = 1, c = 0.3391570949344217123793275284038135702369824934927187... . - Vaclav Kotesovec, Aug 22 2014

A332309 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 3.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 4, 9, 9, 6, 15, 17, 38, 29, 53, 70, 65, 91, 150, 229, 277, 236, 439, 489, 514, 897, 993, 1632, 1521, 2339, 2972, 3257, 4121, 5992, 5303, 7729, 10932, 15157, 17653, 18398, 26305, 31683, 34408, 51885, 58173, 61098, 90519, 101249, 143402, 156905

Offset: 0

Ilya Gutkovskiy, Feb 09 2020

nonn

			a(9) = 6 because we have [8, 1], [7, 2], [5, 4], [4, 5], [2, 7] and [1, 8].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions

Cf. A000726, A003105, A032020, A032021, A332310, A332311.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 09 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!, Sum[b[n - i j, i - 1, p + j], {j, 0, Min[Mod[i, 3], 1, n/i]}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 55] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz *)

A332311 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 5.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 9, 11, 11, 19, 44, 31, 61, 87, 117, 144, 279, 311, 389, 541, 640, 1003, 1225, 2145, 2493, 3452, 3507, 5417, 6671, 8821, 11580, 17959, 21043, 26289, 34797, 41536, 59637, 72707, 85871, 110947, 172472, 175873, 249691, 327801, 418779, 512748

Offset: 0

Ilya Gutkovskiy, Feb 09 2020

nonn

			a(6) = 9 because we have [6], [4, 2], [3, 2, 1], [3, 1, 2], [2, 4], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions

Cf. A032020, A032021, A035959, A096938, A332309, A332310.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 09 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!, Sum[b[n - i j, i - 1, p + j], {j, 0, Min[Mod[i, 5], 1, n/i]}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 55] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

A339086 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 5.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 0, 1, 4, 6, 0, 0, 1, 4, 6, 0, 0, 1, 6, 12, 0, 0, 1, 6, 18, 24, 0, 1, 8, 24, 24, 0, 1, 8, 30, 48, 0, 1, 10, 42, 72, 0, 1, 10, 48, 120, 120, 1, 12, 60, 144, 120, 1, 12, 72, 216, 240, 1, 14, 84, 264, 360, 1, 14, 96, 360, 600, 1, 16, 114

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(18) = 6 because we have [11, 6, 1], [11, 1, 6], [6, 11, 1], [6, 1, 11], [1, 11, 6] and [1, 6, 11].

Index entries for sequences related to compositions

Cf. A003520, A016861, A032020, A032021, A109697, A280454, A337547, A337548, A339059, A339060, A339087, A339088, A339089.

nmax = 78; CoefficientList[Series[Sum[k! x^(k (5 k - 3)/2)/Product[1 - x^(5 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(5*k - 3)/2) / Product_{j=1..k} (1 - x^(5*j)).

A339087 Number of compositions (ordered partitions) of n into distinct parts congruent to 4 mod 5.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 1, 0, 0, 0, 4, 1, 0, 0, 6, 4, 1, 0, 0, 6, 6, 1, 0, 0, 12, 6, 1, 0, 0, 18, 8, 1, 0, 24, 24, 8, 1, 0, 24, 30, 10, 1, 0, 48, 42, 10, 1, 0, 72, 48, 12, 1, 0, 120, 60, 12, 1, 120, 144, 72, 14, 1, 120, 216, 84, 14, 1, 240

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(27) = 6 because we have [14, 9, 4], [14, 4, 9], [9, 14, 4], [9, 4, 14], [4, 14, 9] and [4, 9, 14].

Index entries for sequences related to compositions

Cf. A016897, A017827, A032020, A032021, A109700, A281243, A337547, A337548, A339059, A339060, A339086, A339088, A339089.

nmax = 80; CoefficientList[Series[Sum[k! x^(k (5 k + 3)/2)/Product[1 - x^(5 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(5*k + 3)/2) / Product_{j=1..k} (1 - x^(5*j)).

A339088 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 6.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1, 4, 6, 0, 0, 0, 1, 4, 6, 0, 0, 0, 1, 6, 12, 0, 0, 0, 1, 6, 18, 24, 0, 0, 1, 8, 24, 24, 0, 0, 1, 8, 30, 48, 0, 0, 1, 10, 42, 72, 0, 0, 1, 10, 48, 120, 120, 0, 1, 12, 60, 144, 120, 0, 1, 12, 72, 216, 240, 0, 1, 14, 84, 264, 360

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(21) = 6 because we have [13, 7, 1], [13, 1, 7], [7, 13, 1], [7, 1, 13], [1, 13, 7] and [1, 7, 13].

Index entries for sequences related to compositions

Cf. A005708, A016921, A032020, A032021, A109701, A280456, A337547, A337548, A339059, A339060, A339086, A339087, A339089.

nmax = 83; CoefficientList[Series[Sum[k! x^(k (3 k - 2))/Product[1 - x^(6 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(3*k - 2)) / Product_{j=1..k} (1 - x^(6*j)).

A339089 Number of compositions (ordered partitions) of n into distinct parts congruent to 5 mod 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 4, 1, 0, 0, 0, 6, 4, 1, 0, 0, 0, 6, 6, 1, 0, 0, 0, 12, 6, 1, 0, 0, 0, 18, 8, 1, 0, 0, 24, 24, 8, 1, 0, 0, 24, 30, 10, 1, 0, 0, 48, 42, 10, 1, 0, 0, 72, 48, 12, 1, 0, 0, 120, 60, 12, 1, 0, 120, 144

Offset: 0

Ilya Gutkovskiy, Nov 23 2020

nonn

			a(33) = 6 because we have [17, 11, 5], [17, 5, 11], [11, 17, 5], [11, 5, 17], [5, 17, 11] and [5, 11, 17].

Index entries for sequences related to compositions

Cf. A016969, A017837, A032020, A032021, A109702, A281244, A337547, A337548, A339059, A339060, A339086, A339087, A339088.

nmax = 86; CoefficientList[Series[Sum[k! x^(k (3 k + 2))/Product[1 - x^(6 j), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} k! * x^(k*(3*k + 2)) / Product_{j=1..k} (1 - x^(6*j)).

cp's OEIS Frontend

A331846 Number of compositions (ordered partitions) of n into distinct squarefree parts.

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A331847 Number of compositions (ordered partitions) of n into distinct prime powers (1 excluded).

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A332032 G.f.: Sum_{k>=1} k! * x^(k^2) / (1 - x^(2*k)).

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A218694 Carlitz compositions of n into odd parts.

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A332309 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 3.

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A332311 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 5.

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A339086 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 5.

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A339087 Number of compositions (ordered partitions) of n into distinct parts congruent to 4 mod 5.

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A339088 Number of compositions (ordered partitions) of n into distinct parts congruent to 1 mod 6.