A053251 -id:A053251 - cp's OEIS frontend

A357705 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 3, 2, 3, 2, 0, 1, 0, 4, 2, 4, 1, 3, 0, 1, 0, 4, 3, 3, 6, 2, 3, 0, 1, 0, 5, 3, 5, 3, 7, 2, 4, 0, 1, 0, 5, 4, 5, 4, 9, 7, 3, 4, 0, 1, 0, 6, 4, 7, 3, 12, 5, 10, 3, 5, 0, 1

Offset: 0

Gus Wiseman, Oct 10 2022

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ...

			Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  2  2  0  1
  0  3  1  2  0  1
  0  3  2  3  2  0  1
  0  4  2  4  1  3  0  1
  0  4  3  3  6  2  3  0  1
  0  5  3  5  3  7  2  4  0  1
  0  5  4  5  4  9  7  3  4  0  1
  0  6  4  7  3 12  5 10  3  5  0  1
  0  6  5  7  5 10 16  7 11  4  5  0  1
  0  7  5  9  5 14 11 18  7 14  4  6  0  1
Row n = 7 counts the following reversed partitions:
  .  (16)   (25)   (34)       (1123)  (1114)   .  (7)
     (115)  (223)  (1222)             (11113)
     (124)         (111112)           (11122)
     (133)         (1111111)

Row sums are A000041.
First nonzero entry of each row is A004526.
The central column is A357640, half A357639.
For original alternating sum we have A344651, ordered A097805.
The half-alternating version is A357704.
The ordered non-reverse version (compositions) is A357646, half A357645.
The non-reverse version is A357638, half A357637.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew  A357634.
Cf. A035363, A035594, A053251, A298311, A357136, A357189, A357487, A357488, A357624, A357631, A357632, A357636.

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]),{i,Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n],skats[#]==k&]],{n,0,11},{k,-n,n,2}]

A192433 Coefficients of a mock theta function.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 9, 12, 16, 19, 24, 31, 37, 46, 57, 68, 83, 101, 120, 143, 171, 202, 239, 283, 331, 388, 455, 529, 616, 716, 827, 957, 1105, 1270, 1460, 1676, 1918, 2193, 2506, 2854, 3248, 3695, 4191, 4752, 5382, 6082, 6870, 7752, 8732, 9829

Offset: 0

Jeremy Lovejoy, Jun 30 2011

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
K. Bringmann, K. Hikami and J. Lovejoy, On the modularity of the unified WRT invariants of certain Seifert manifolds
K. Bringmann, K. Hikami, and J. Lovejoy, On the modularity of the unified WRT invariants of certain Seifert manifolds, Adv. Appl. Math. 46 (2011), 86-93.

Cf. A053251.

nmax = 60; CoefficientList[Series[Sum[x^k*Product[1 + x^j, {j, 1, 2*k}], {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 16 2025 *)

N=66; q='q+O('q^N); gf=sum(n=0,N,q^n*prod(k=1,2*n,1+q^k)); Vec(gf) \\ Joerg Arndt, Jul 01 2011

a(n) = A053251(2*n+1).

a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(2*n)). - Vaclav Kotesovec, Jun 12 2019

A356843 Numbers k such that the k-th composition in standard order covers an interval of positive integers (gapless) but contains no 1's.

Original entry on oeis.org

2, 4, 8, 10, 16, 18, 20, 32, 36, 42, 64, 68, 72, 74, 82, 84, 128, 136, 146, 148, 164, 170, 256, 264, 272, 274, 276, 290, 292, 296, 298, 324, 328, 330, 338, 340, 512, 528, 548, 580, 584, 586, 594, 596, 658, 660, 676, 682, 1024, 1040, 1056, 1092, 1096, 1098

Offset: 1

Gus Wiseman, Sep 01 2022

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.

			The terms together with their corresponding standard compositions begin:
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   18: (3,2)
   20: (2,3)
   32: (6)
   36: (3,3)
   42: (2,2,2)
   64: (7)
   68: (4,3)
   72: (3,4)
   74: (3,2,2)
   82: (2,3,2)
   84: (2,2,3)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
A subset of A022340.
These compositions are counted by A251729.
The unordered version (using Heinz numbers of partitions) is A356845.
A333217 ranks complete compositions.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356841 ranks gapless compositions, counted by A107428.
A356842 ranks non-gapless compositions, counted by A356846.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073491, A073492, A073493, A137921, A356224/A356225.

nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],!MemberQ[stc[#],1]&&nogapQ[stc[#]]&]

Complement of A333217 in A356841.

A132969 Expansion of phi(q) * chi(q) in powers of q where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 2, 1, 5, 5, 3, 5, 6, 10, 10, 8, 13, 15, 15, 16, 23, 27, 25, 30, 35, 40, 42, 45, 55, 66, 68, 70, 86, 95, 100, 110, 125, 141, 150, 161, 185, 207, 215, 235, 266, 293, 310, 335, 375, 410, 438, 470, 521, 575, 610, 653, 725, 785, 835, 900, 983, 1070, 1140

Offset: 0

Michael Somos, Sep 04 2007

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 3*x + 2*x^2 + x^3 + 5*x^4 + 5*x^5 + 3*x^6 + 5*x^7 + 6*x^8 + 10*x^9 + ...
G.f. = 1/q + 3*q^23 + 2*q^47 + q^71 + 5*q^95 + 5*q^119 + 3*q^143 + 5*q^167 +...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; top of p. 60.

G. C. Greubel, Table of n, a(n) for n = 0..1000
George E. Andrews, Integer partitions with even parts below odd parts and the mock theta functions, to appear in Annals of Combinatorics, 2017.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053250, A053251, A124226, A132970.

nmax = 60; CoefficientList[Series[Product[(1 - x^(2*k)) * ( (1 + x^k) / (1 + x^(2*k)) )^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Oct 31 2015 *)

{a(n) = if( n<0, 0, polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + x*O(x^n)) * sum(k=1, sqrtint(n), 2 * x^k^2, 1), n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A) * eta(x^4 + A))^3, n))};

Expansion of phi(q) + 2 * psi(q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.
Expansion of q^(1/24) * eta(q^2)^7 / (eta(q) * eta(q^4))^3 in powers of q.
Euler transform of period 4 sequence [ 3, -4, 3, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t).
G.f.: ( Sum_{k in Z} x^k^2 ) * ( Product_{k>0} (1 + x^(2*k-1)) ).
G.f.: Product_{k>0} (1 - x^(2*k)) * ((1 + x^k) / (1 + x^(2*k)))^3.
a(n) = (-1)^n * A132970(n). a(n) = (-1)^n * A124226(n) unless n=1.
a(n) ~ exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Sep 08 2015

A239327 Number of palindromic Carlitz compositions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 5, 5, 7, 10, 14, 14, 25, 26, 42, 48, 75, 79, 132, 142, 226, 252, 399, 432, 704, 760, 1223, 1336, 2143, 2328, 3759, 4079, 6564, 7150, 11495, 12496, 20135, 21874, 35215, 38310, 61639, 67018, 107912, 117298, 188839, 205346, 330515, 359350, 578525, 628951

Offset: 0

Geoffrey Critzer, Mar 16 2014

nonn

A palindromic composition is a composition that is identical to its own reverse. There are 2^floor(n/2) palindromic compositions. A Carlitz composition has no two consecutive equal parts (A003242). This sequence enumerates compositions that are both palindromic and Carlitz.

Also the number of odd-length integer compositions of n into parts that are alternately unequal and equal (n > 0). The unordered version (partitions) is A053251. - Gus Wiseman, Feb 26 2022

			a(9) = 7 because we have: 9, 1+7+1, 2+5+2, 4+1+4, 1+3+1+3+1, 2+1+3+1+2, 1+2+3+2+1. 2+3+4 is not counted because it is not palindromic. 3+3+3 is not counted because it has consecutive equal parts.

S. Heubach and T. Mansour, Compositions of n with parts in a set, Congr. Numer. 168 (2004), 127-143.
S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2010, page 67.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Petros Hadjicostas, Cyclic, Dihedral and Symmetrical Carlitz Compositions of a Positive Integer, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.5.

Carlitz compositions are counted by A003242.
Palindromic compositions are counted by A016116.
The unimodal case is A096441.
Cf. A053251, A122129, A122130, A351003, A351006, A351007.

b:= proc(n, i) option remember; `if`(i=0, 0, `if`(n=0, 1,
      add(`if`(i=j, 0, b(n-j, j)), j=1..n)))
    end:
a:= n-> `if`(n=0, 1, add(b(i, n-2*i), i=0..n/2)):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 16 2014

nn=50;CoefficientList[Series[(1+Sum[x^j(1-x^j)/(1+x^(2j)),{j,1,nn}])/(1-Sum[x^(2j)/(1+x^(2j)),{j,1,nn}]),{x,0,nn}],x]
(* or *)
Table[Length[Select[Level[Map[Permutations,Partitions[n]],{2}],Apply[And,Table[#[[i]]==#[[Length[#]-i+1]],{i,1,Floor[Length[#]/2]}]]&&Apply[And,Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1}]]&]],{n,0,20}]

a(n) = polcoeff((1 + sum(j=1, n, x^j*(1-x^j)/(1+x^(2*j)) + O(x*x^n))) / (1 - sum(j=1, n, x^(2*j)/(1+x^(2*j)) + O(x*x^n))), n); \\ Andrew Howroyd, Oct 12 2017

G.f.: (1 + Sum_{j>=1} x^j*(1-x^j)/(1+x^(2*j))) / (1 - Sum_{j>=1} x^(2*j)/(1+x^(2*j))).

a(n) ~ c / r^n, where r = 0.7558768372943356987836792261127971643747976345582722756032673... is the root of the equation sum_{j>=1} x^(2*j)/(1+x^(2*j)) = 1, c = 0.5262391407444644722747255167331403939384758635340487280277... if n is even and c = 0.64032989654153238794063877354074732669441634551692765196197... if n is odd. - Vaclav Kotesovec, Aug 22 2014

A351595 Number of odd-length integer partitions y of n such that y_i > y_{i+1} for all even i.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 16, 20, 24, 30, 35, 44, 52, 63, 74, 90, 105, 126, 148, 175, 204, 242, 280, 330, 382, 446, 515, 600, 690, 800, 919, 1060, 1214, 1398, 1595, 1830, 2086, 2384, 2711, 3092, 3506, 3988, 4516, 5122, 5788, 6552, 7388, 8345

Offset: 0

Gus Wiseman, Feb 25 2022

nonn

			The a(1) = 1 through a(12) = 10 partitions (A..C = 10..12):
  1   2   3   4   5     6     7     8     9     A     B       C
                  221   321   331   332   432   442   443     543
                              421   431   441   532   542     552
                                    521   531   541   551     642
                                          621   631   632     651
                                                721   641     732
                                                      731     741
                                                      821     831
                                                      33221   921
                                                              43221

The ordered version (compositions) is A000213 shifted right once.
All odd-length partitions are counted by A027193.
The opposite appears to be A122130, even-length A351008, any length A122129.
This appears to be the odd-length case of A122135, even-length A122134.
The case that is constant at odd indices:
- any length: A351005
- odd length: A351593
- even length: A035457
- opposite any length: A351006
- opposite odd length: A053251
- opposite even length: A351007
For equality instead of inequality:
- any length: A351003
- odd-length: A000009 (except at 0)
- even-length: A351012
- opposite any length: A351004
- opposite odd-length: A351594
- opposite even-length: A035363
Cf. A000041, A000070, A003242, A027383, A236559, A236914, A350842.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]>#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]

A356842 Numbers k such that the k-th composition in standard order does not cover an interval of positive integers (not gapless).

Original entry on oeis.org

9, 12, 17, 19, 24, 25, 28, 33, 34, 35, 39, 40, 48, 49, 51, 56, 57, 60, 65, 66, 67, 69, 70, 71, 73, 76, 79, 80, 81, 88, 96, 97, 98, 99, 100, 103, 104, 112, 113, 115, 120, 121, 124, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 144, 145

Offset: 1

Gus Wiseman, Sep 01 2022

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.

			The terms and their corresponding standard compositions begin:
   9: (3,1)
  12: (1,3)
  17: (4,1)
  19: (3,1,1)
  24: (1,4)
  25: (1,3,1)
  28: (1,1,3)
  33: (5,1)
  34: (4,2)
  35: (4,1,1)
  39: (3,1,1,1)
  40: (2,4)
  48: (1,5)
  49: (1,4,1)
  51: (1,3,1,1)
  56: (1,1,4)
  57: (1,1,3,1)
  60: (1,1,1,3)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
An unordered version is A073492, complement A073491.
These compositions are counted by the complement of A107428.
The complement is A356841.
The gapless but non-initial version is A356843, unordered A356845.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A333217, A356224/A356225.

nogapQ[m_]:=m=={}||Union[m]==Range[Min[m],Max[m]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!nogapQ[stc[#]]&]

A304705 Number of partitions (d1,d2,...,dm) of n such that d1/1 >= d2/2 >= ... >= dm/m and 0 < d1 <= d2 <= ... <= dm.

Original entry on oeis.org

1, 1, 2, 3, 3, 4, 6, 5, 6, 8, 9, 9, 12, 11, 14, 17, 16, 17, 23, 22, 27, 31, 30, 33, 40, 41, 46, 50, 54, 57, 70, 70, 77, 88, 92, 99, 111, 115, 129, 142, 152, 160, 175, 183, 199, 223, 234, 255, 283, 299, 328, 347, 370, 390, 430, 455, 489, 523, 557, 592, 642, 674, 724, 784

Offset: 0

Seiichi Manyama, May 17 2018

nonn

			n | Partition (d1,d2,...,dm)    | (d1/1, d2/2, ... , dm/m)
--+-----------------------------+---------------------------------------------
1 | (1)                         | (1)
2 | (2)                         | (2)
  | (1, 1)                      | (1, 1/2)
3 | (3)                         | (3)
  | (1, 2)                      | (1, 1)
  | (1, 1, 1)                   | (1, 1/2, 1/3)
4 | (4)                         | (4)
  | (2, 2)                      | (2, 1)
  | (1, 1, 1, 1)                | (1, 1/2, 1/3, 1/4)
5 | (5)                         | (5)
  | (2, 3)                      | (2, 3/2)
  | (1, 2, 2)                   | (1, 1, 2/3)
  | (1, 1, 1, 1, 1)             | (1, 1/2, 1/3, 1/4, 1/5)
6 | (6)                         | (6)
  | (2, 4)                      | (2, 2)
  | (3, 3)                      | (3, 3/2)
  | (1, 2, 3)                   | (1, 1, 1)
  | (2, 2, 2)                   | (2, 1, 2/3)
  | (1, 1, 1, 1, 1, 1)          | (1, 1/2, 1/3, 1/4, 1/5, 1/6)
7 | (7)                         | (7)
  | (3, 4)                      | (3, 2)
  | (2, 2, 3)                   | (2, 1, 1)
  | (1, 2, 2, 2)                | (1, 1, 2/3, 1/2)
  | (1, 1, 1, 1, 1, 1, 1)       | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7)
8 | (8)                         | (8)
  | (3, 5)                      | (3, 5/2)
  | (4, 4)                      | (4, 2/1)
  | (2, 3, 3)                   | (2, 3/2, 1)
  | (2, 2, 2, 2)                | (2, 1, 2/3, 1/2)
  | (1, 1, 1, 1, 1, 1, 1, 1)    | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8)
9 | (9)                         | (9)
  | (3, 6)                      | (3, 3)
  | (4, 5)                      | (4, 5/2)
  | (2, 3, 4)                   | (2, 3/2, 4/3)
  | (3, 3, 3)                   | (3, 3/2, 1)
  | (1, 2, 3, 3)                | (1, 1, 1, 3/4)
  | (1, 2, 2, 2, 2)             | (1, 1, 2/3, 1/2, 2/5)
  | (1, 1, 1, 1, 1, 1, 1, 1, 1) | (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9)

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A053251, A053282, A304706, A304707, A304708.

b:= proc(n, r, i, t) option remember; `if`(n=0, 1, `if`(i>n, 0,
      b(n, r, i+1, t)+`if`(i/t>r, 0, b(n-i, i/t, i, t+1))))
    end:
a:= n-> b(n$2, 1$2):
seq(a(n), n=0..80);  # Alois P. Heinz, May 17 2018

b[n_, r_, i_, t_] := b[n, r, i, t] = If[n == 0, 1, If[i > n, 0, b[n, r, i + 1, t] + If[i/t > r, 0, b[n - i, i/t, i, t + 1]]]];
a[n_] := b[n, n, 1, 1];
a /@ Range[0, 80] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 5, 4, 6, 4, 8, 6, 9, 6, 12, 7, 14, 10, 16, 11, 20, 13, 24, 16, 28, 18, 34, 21, 40, 26, 46, 30, 56, 34, 64, 41, 75, 48, 88, 54, 102, 64, 118, 73, 138, 84, 159, 98, 182, 112, 210, 128, 242, 148, 276, 168, 318

Offset: 0

Gus Wiseman, Feb 23 2022

nonn

Also odd-length partitions whose run-lengths are all 2's, except for the last, which is 1.

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3  4  5    6  7    8    9    A    B      C    D      E    F
              221     331  332  441  442  443    552  553    554  663
                                          551         661    662  771
                                          33221       44221       44331
                                                                  55221

The even-length ordered version is A003242, ranked by A351010.
The opposite version is A053251, even-length A351007, any length A351006.
This is the odd-length case of A351005, even-length A035457.
With only equalities we get:
  - opposite any length: A351003
  - opposite odd-length: A000009  (except at 0)
  - opposite even-length: A351012
- any length: A351004
- odd-length: A351594
- even-length: A035363
Without equalities we get:
  - opposite any length: A122129 (apparently)
  - opposite odd-length: A122130 (apparently)
  - opposite even-length: A351008
- any length: A122135 (apparently)
- odd-length: A351595
- even-length: A122134 (apparently)
Cf. A000070, A000984, A027383, A053738, A236559, A236914, A350842, A350844.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[If[EvenQ[i],#[[i]]!=#[[i+1]],#[[i]]==#[[i+1]]],{i,Length[#]-1}]&]],{n,0,30}]

A356604 Number of integer compositions of n into odd parts covering an initial interval of odd positive integers.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 5, 9, 13, 24, 40, 61, 101, 160, 257, 415, 679, 1103, 1774, 2884, 4656, 7517, 12165, 19653, 31753, 51390, 83134, 134412, 217505, 351814, 569081, 920769, 1489587, 2409992, 3899347, 6309059, 10208628, 16518910, 26729830, 43254212, 69994082

Offset: 0

Gus Wiseman, Aug 30 2022

nonn

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (11)  (111)  (13)    (113)    (1113)    (133)      (1133)
                    (31)    (131)    (1131)    (313)      (1313)
                    (1111)  (311)    (1311)    (331)      (1331)
                            (11111)  (3111)    (11113)    (3113)
                                     (111111)  (11131)    (3131)
                                               (11311)    (3311)
                                               (13111)    (111113)
                                               (31111)    (111131)
                                               (1111111)  (111311)
                                                          (113111)
                                                          (131111)
                                                          (311111)
                                                          (11111111)
The a(9) = 24 compositions:
  (135)  (11133)  (1111113)  (111111111)
  (153)  (11313)  (1111131)
  (315)  (11331)  (1111311)
  (351)  (13113)  (1113111)
  (513)  (13131)  (1131111)
  (531)  (13311)  (1311111)
         (31113)  (3111111)
         (31131)
         (31311)
         (33111)

The case of partitions is A053251, ranked by A356232 and A356603.
These compositions are ranked by the intersection of A060142 and A333217.
This is the odd initial case of A107428.
This is the odd restriction of A107429.
This is the normal/covering case of A324969 (essentially A000045).
The non-initial version is A356605.
A000041 counts partitions, compositions A011782.
A055932 lists numbers with prime indices covering an initial interval.
A066208 lists numbers with all odd prime indices, counted by A000009.
Cf. A001221, A001222, A001227, A005408, A061395, A066205, A073493, A137921, A356224.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[(#+1)/2]&]],{n,0,15}]

More terms from Alois P. Heinz, Sep 01 2022

cp's OEIS Frontend

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