A332304 -id:A332304 - cp's OEIS frontend

A027193 Number of partitions of n into an odd number of parts.

0, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 29, 37, 52, 66, 90, 113, 151, 190, 248, 310, 400, 497, 632, 782, 985, 1212, 1512, 1851, 2291, 2793, 3431, 4163, 5084, 6142, 7456, 8972, 10836, 12989, 15613, 18646, 22316, 26561, 31659, 37556, 44601, 52743, 62416, 73593, 86809, 102064, 120025, 140736

Offset: 0

Clark Kimberling

nonn

Number of partitions of n in which greatest part is odd.
Number of partitions of n+1 into an even number of parts, the least being 1. Example: a(5)=4 because we have [5,1], [3,1,1,1], [2,1,1] and [1,1,1,1,1,1].
Also number of partitions of n+1 such that the largest part is even and occurs only once. Example: a(5)=4 because we have [6], [4,2], [4,1,1] and [2,1,1,1,1]. - Emeric Deutsch, Apr 05 2006
Also the number of partitions of n such that the number of odd parts and the number of even parts have opposite parities.  Example: a(8)=10 is a count of these partitions: 8, 611, 521, 431, 422, 41111, 332, 32111, 22211, 2111111. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021
In Chaves 2011 see page 38 equation (3.20). - Michael Somos, Dec 28 2014
Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1).  When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k).  Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n.  Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts.  That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms.  The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

			G.f. = x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 8*x^7 + 10*x^8 + 16*x^9 + 20*x^10 + ...
From _Gus Wiseman_, Feb 11 2021: (Start)
The a(1) = 1 through a(8) = 10 partitions into an odd number of parts are the following. The Heinz numbers of these partitions are given by A026424.
  (1)  (2)  (3)    (4)    (5)      (6)      (7)        (8)
            (111)  (211)  (221)    (222)    (322)      (332)
                          (311)    (321)    (331)      (422)
                          (11111)  (411)    (421)      (431)
                                   (21111)  (511)      (521)
                                            (22111)    (611)
                                            (31111)    (22211)
                                            (1111111)  (32111)
                                                       (41111)
                                                       (2111111)
The a(1) = 1 through a(8) = 10 partitions whose greatest part is odd are the following. The Heinz numbers of these partitions are given by A244991.
  (1)  (11)  (3)    (31)    (5)      (33)      (7)        (53)
             (111)  (1111)  (32)     (51)      (52)       (71)
                            (311)    (321)     (322)      (332)
                            (11111)  (3111)    (331)      (521)
                                     (111111)  (511)      (3221)
                                               (3211)     (3311)
                                               (31111)    (5111)
                                               (1111111)  (32111)
                                                          (311111)
                                                          (11111111)
(End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 39, Example 7.

T. D. Noe, Table of n, a(n) for n = 0..999
Roland Bacher and P. De La Harpe, Conjugacy growth series of some infinitely generated groups, hal-01285685v2, 2016.
D. R. C. Chaves, Um estudo combinatório e comparativo de identidades teta parciais de Andrews e Ramanujan, 2011. In Portuguese.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_0(n).

Cf. A000041, A000700, A000701, A026837, A046682.
The Heinz numbers of these partitions are A026424 or A244991.
The even-length version is A027187.
The case of odd sum as well as length is A160786, ranked by A340931.
The case of odd maximum as well as length is A340385.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A067659 counts strict partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
A000009 counts partitions into odd parts, ranked by A066208.
A026804 counts partitions whose least part is odd.
A058695 counts partitions of odd numbers, ranked by A300063.
A072233 counts partitions by sum and length.
A101707 counts partitions of odd positive rank.
Cf. A078408, A174725, A236914, A300272, A340831.

g:=sum(x^(2*k)/product(1-x^j,j=1..2*k-1),k=1..40): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=1..45); # Emeric Deutsch, Apr 05 2006

nn=40;CoefficientList[Series[ Sum[x^(2j+1)Product[1/(1- x^i),{i,1,2j+1}],{j,0,nn}],{x,0,nn}],x]  (* Geoffrey Critzer, Dec 01 2012 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n], OddQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n], OddQ[ First@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[ n + 1], #[[-1]] == 1 && EvenQ[ Length@#] &]]; (* Michael Somos, Dec 28 2014 *)
a[ n_] := If[ n < 1, 0, Length@Select[ IntegerPartitions[ n + 1], EvenQ[ First@#] && (Length[#] < 2 || #[[1]] != #[[2]]) &]]; (* Michael Somos, Dec 28 2014 *)

{a(n) = if( n<1, 0, polcoeff( sum( k=1, n, if( k%2, x^k / prod( j=1, k, 1 - x^j, 1 + x * O(x^(n-k)) ))), n))}; /* Michael Somos, Jul 24 2012 */

q='q+O('q^66); concat([0], Vec( (1/eta(q)-eta(q)/eta(q^2))/2 ) ) \\ Joerg Arndt, Mar 23 2014

a(n) = (A000041(n) - (-1)^n*A000700(n)) / 2.
For g.f. see under A027187.
G.f.: Sum(k>=1, x^(2*k-1)/Product(j=1..2*k-1, 1-x^j ) ). - Emeric Deutsch, Apr 05 2006
G.f.: - Sum(k>=1, (-x)^(k^2)) / Product(k>=1, 1-x^k ). - Joerg Arndt, Feb 02 2014
G.f.: Sum(k>=1, x^(k*(2*k-1)) / Product(j=1..2*k, 1-x^j)). - Michael Somos, Dec 28 2014
a(2*n) = A000701(2*n), a(2*n-1) = A046682(2*n-1); a(n) = A000041(n)-A027187(n). - Reinhard Zumkeller, Apr 22 2006

A067659 Number of partitions of n into distinct parts such that number of parts is odd.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 82, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2048, 2291, 2560, 2859, 3189, 3554, 3958, 4404

Offset: 0

Naohiro Nomoto, Feb 23 2002

Ramanujan theta functions: phi(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			From _Gus Wiseman_, Jan 09 2021: (Start)
The a(5) = 1 through a(15) = 14 partitions (A-F = 10..15):
  5   6     7     8     9     A     B     C     D     E     F
      321   421   431   432   532   542   543   643   653   654
                  521   531   541   632   642   652   743   753
                        621   631   641   651   742   752   762
                              721   731   732   751   761   843
                                    821   741   832   842   852
                                          831   841   851   861
                                          921   931   932   942
                                                A21   941   951
                                                      A31   A32
                                                      B21   A41
                                                            B31
                                                            C21
                                                            54321
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Joerg Arndt, Matters Computational (The Fxtbook), end of section 16.4.2 "Partitions into distinct parts", pp.348ff
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160 (March 2016), Pages 60-75, function q_o(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Dominates A000009.
Numbers with these strict partitions as binary indices are A000069.
The non-strict version is A027193.
The Heinz numbers of these partitions are A030059.
The even version is A067661.
The version for rank is A117193, with non-strict version A101707.
The ordered version is A332304, with non-strict version A166444.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A174726 counts ordered factorizations of odd length.
- A339890 counts factorizations of odd length.
A008289 counts strict partitions by sum and length.
A026804 counts partitions whose least part is odd, with strict case A026832.
Cf. A000700, A027187, A030229, A117192, A332305.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2, 0, If[n == 0, t, Sum[b[n - i*j, i - 1, Abs[t - j]], {j, 0, Min[n/i, 1]}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
CoefficientList[Normal[Series[(QPochhammer[-x, x]-QPochhammer[x])/2, {x, 0, 100}]], x] (* Andrey Zabolotskiy, Apr 12 2017 *)
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&OddQ[Length[#]]&]],{n,0,30}] (* Gus Wiseman, Jan 09 2021 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)/eta(x+A) - eta(x+A))/2, n))} /* Michael Somos, Feb 14 2006 */

N=66;  q='q+O('q^N);  S=1+2*sqrtint(N);
gf=sum(n=1,S, (n%2!=0) * q^(n*(n+1)/2) / prod(k=1,n, 1-q^k ) );
concat( [0], Vec(gf) )  /* Joerg Arndt, Oct 20 2012 */

N=66;  q='q+O('q^N);  S=1+sqrtint(N);
gf=sum(n=1, S, q^(2*n^2-n) / prod(k=1, 2*n-1, 1-q^k ) );
concat( [0], Vec(gf) )  \\ Joerg Arndt, Apr 01 2014

For g.f. see under A067661.
a(n) = (A000009(n)-A010815(n))/2. - Vladeta Jovovic, Feb 24 2002
Expansion of (1-phi(-q))/(2*chi(-q)) in powers of q where phi(),chi() are Ramanujan theta functions. - Michael Somos, Feb 14 2006
G.f.: sum(n>=1, q^(2*n^2-n) / prod(k=1..2*n-1, 1-q^k ) ). [Joerg Arndt, Apr 01 2014]
a(n) = A067661(n) - A010815(n). - Andrey Zabolotskiy, Apr 12 2017
A000009(n) = a(n) + A067661(n). - Gus Wiseman, Jan 09 2021

A174726 a(n) = (A002033(n-1) - A008683(n))/2.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 2, 4, 1, 7, 1, 8, 1, 1, 1, 13, 1, 1, 1, 10, 1, 7, 1, 4, 4, 1, 1, 24, 1, 4, 1, 4, 1, 10, 1, 10, 1, 1, 1, 22, 1, 1, 4, 16, 1, 7, 1, 4, 1, 7, 1, 38, 1, 1, 4, 4, 1

Offset: 1

Mats Granvik, Mar 28 2010

nonn

a(n) is the number of permutation matrices with a negative contribution to the determinant that is the Möbius function. See A174725 for how the determinant is defined. - Mats Granvik, May 26 2017
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an odd number of factors > 1. The unordered case is A339890. For example, the a(n) factorizations for n = 8, 12, 24, 30, 32, 36 are:
  (8)      (12)     (24)     (30)     (32)         (36)
  (2*2*2)  (2*2*3)  (2*2*6)  (2*3*5)  (2*2*8)      (2*2*9)
           (2*3*2)  (2*3*4)  (2*5*3)  (2*4*4)      (2*3*6)
           (3*2*2)  (2*4*3)  (3*2*5)  (2*8*2)      (2*6*3)
                    (2*6*2)  (3*5*2)  (4*2*4)      (2*9*2)
                    (3*2*4)  (5*2*3)  (4*4*2)      (3*2*6)
                    (3*4*2)  (5*3*2)  (8*2*2)      (3*3*4)
                    (4*2*3)           (2*2*2*2*2)  (3*4*3)
                    (4*3*2)                        (3*6*2)
                    (6*2*2)                        (4*3*3)
                                                   (6*2*3)
                                                   (6*3*2)
                                                   (9*2*2)
(End)

Mats Granvik, Table of n, a(n) for n = 1..10000

The even version is A174725.
The unordered case is A339890, with even version A339846.
A001055 counts factorizations, with strict case A045778.
A074206 counts ordered factorizations, with strict case A254578.
A251683 counts ordered factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A027193 counts partitions of odd length.
- A067659 counts strict partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A166444 counts compositions of odd length.
- A332304 counts strict compositions of odd length.
Cf. A002033, A024430, A027187, A050320, A052841, A058695, A160786, A316439.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],OddQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)

a(n) = (A002033(n-1) - A008683(n))/2. - Mats Granvik, May 26 2017

For n > 0, a(n) + A174725(n) = A074206(n). - Gus Wiseman, Jan 04 2021

A340855 Numbers that can be factored into factors > 1, the least of which is odd.

Original entry on oeis.org

3, 5, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

These are numbers that are odd or have an odd divisor 1 < d <= n/d.

			The sequence of terms together with their prime indices begins:
     3: {2}          27: {2,2,2}      48: {1,1,1,1,2}
     5: {3}          29: {10}         49: {4,4}
     7: {4}          30: {1,2,3}      50: {1,3,3}
     9: {2,2}        31: {11}         51: {2,7}
    11: {5}          33: {2,5}        53: {16}
    12: {1,1,2}      35: {3,4}        54: {1,2,2,2}
    13: {6}          36: {1,1,2,2}    55: {3,5}
    15: {2,3}        37: {12}         56: {1,1,1,4}
    17: {7}          39: {2,6}        57: {2,8}
    18: {1,2,2}      40: {1,1,1,3}    59: {17}
    19: {8}          41: {13}         60: {1,1,2,3}
    21: {2,4}        42: {1,2,4}      61: {18}
    23: {9}          43: {14}         63: {2,2,4}
    24: {1,1,1,2}    45: {2,2,3}      65: {3,6}
    25: {3,3}        47: {15}         66: {1,2,5}
For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).

The version looking at greatest factor is A057716.
The version for twice-balanced is A340657, with complement A340656.
These factorization are counted by A340832.
The complement is A340854.
A033676 selects the maximum inferior divisor.
A038548 counts inferior divisors, listed by A161906.
A055396 selects the least prime index.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A332304 counts strict compositions of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.

Select[Range[100],Function[n,n>1&&(OddQ[n]||Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]!={})]]

A340385 Number of integer partitions of n into an odd number of parts, the greatest of which is odd.

Original entry on oeis.org

1, 0, 2, 0, 3, 1, 6, 3, 10, 7, 18, 15, 30, 28, 51, 50, 82, 87, 134, 145, 211, 235, 331, 375, 510, 586, 779, 901, 1172, 1366, 1750, 2045, 2581, 3026, 3778, 4433, 5476, 6430, 7878, 9246, 11240, 13189, 15931, 18670, 22417, 26242, 31349, 36646, 43567, 50854

Offset: 1

Gus Wiseman, Jan 08 2021

nonn

			The a(3) = 2 through a(10) = 7 partitions:
  3     5       321   7         332     9           532
  111   311           322       521     333         541
        11111         331       32111   522         721
                      511               531         32221
                      31111             711         33211
                      1111111           32211       52111
                                        33111       3211111
                                        51111
                                        3111111
                                        111111111

Partitions of odd length are counted by A027193, ranked by A026424.
Partitions with odd maximum are counted by A027193, ranked by A244991.
The Heinz numbers of these partitions are given by A340386.
Other cases of odd length:
- A024429 counts set partitions of odd length.
- A067659 counts strict partitions of odd length.
- A089677 counts ordered set partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
A000009 counts partitions into odd parts, ranked by A066208.
A026804 counts partitions whose least part is odd.
A058695 counts partitions of odd numbers, ranked by A300063.
A072233 counts partitions by sum and length.
A101707 counts partitions with odd rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Cf. A000700, A027187, A078408, A174725, A236914.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]*Max[#]]&]],{n,30}]

A332305 Number of compositions (ordered partitions) of n into distinct parts such that number of parts is even.

Original entry on oeis.org

1, 0, 0, 2, 2, 4, 4, 6, 6, 8, 32, 34, 58, 84, 132, 158, 230, 280, 376, 450, 570, 1388, 1556, 2398, 3310, 4920, 6600, 9674, 12122, 16684, 21340, 28110, 34974, 45392, 55208, 69274, 124498, 143676, 204012, 270758, 377966, 493024, 690304, 895434, 1223826, 1562948

Offset: 0

Ilya Gutkovskiy, Feb 09 2020

nonn

			a(5) = 4 because we have [4, 1], [3, 2], [2, 3] and [1, 4].

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

Cf. A027187, A032020, A034008, A067661, A332304.

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

nmax = 45; CoefficientList[Series[Sum[(2 k)! x^(k (2 k + 1))/Product[1 - x^j, {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]

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

a(n) = A032020(n) - A332304(n).

A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.

Original entry on oeis.org

0, 1, 1, 7, 37, 271, 2341, 23647, 272917, 3543631, 51123781, 811316287, 14045783797, 263429174191, 5320671485221, 115141595488927, 2657827340990677, 65185383514567951, 1692767331628422661, 46400793659664205567, 1338843898122192101557

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004

Stirling transform of A005212(n)=[1,0,6,0,120,0,5040,...] is a(n)=[1,1,7,37,271,...]. - Michael Somos, Mar 04 2004

Occurs also as first column of a matrix-inversion occurring in a sum-of-like-powers problem. Consider the problem for any fixed natural number m>2 of finding solutions to sum(k=1,n,k^m) = (k+1)^m. Erdos conjectured that there are no solutions for n,m>2. Let D be the matrix of differences of D[m,n] := sum(k=1,n,k^m) - (k+1)^m. Then the generating functions for the rows of this matrix D constitute a set of polynomials in n (for varying n along columns) and the m-th polynomial defining the m-th row. Let GF_D be the matrix of the coefficients of this set of polynomials. Then the present sequence is the (unsigned) second column of GF_D^-1. - Gottfried Helms, Apr 01 2007

			From _Gus Wiseman_, Jan 06 2021: (Start)
a(n) is the number of ordered set partitions of {1..n} into an odd number of blocks. The a(1) = 1 through a(3) = 7 ordered set partitions are:
  {{1}}  {{1,2}}  {{1,2,3}}
                  {{1},{2},{3}}
                  {{1},{3},{2}}
                  {{2},{1},{3}}
                  {{2},{3},{1}}
                  {{3},{1},{2}}
                  {{3},{2},{1}}
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J.-C. Aval, V. Féray, J.-C. Novelli, J.-Y. Thibon, Quasi-symmetric functions as polynomial functions on Young diagrams, arXiv preprint arXiv:1312.2727, 2013
Gottfried Helms, Discussion of a problem concerning summing of like powers

Ordered set partitions are counted by A000670.
The case of (unordered) set partitions is A024429.
The complement (even-length ordered set partitions) is counted by A052841.
A058695 counts partitions of odd numbers, ranked by A300063.
A101707 counts partitions of odd positive rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340102 counts odd-length factorizations into odd factors.
A340692 counts partitions of odd rank.
Other cases of odd length:
- A027193 counts partitions of odd length.
- A067659 counts strict partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
Cf. A000700, A026424, A027187, A028260, A078408, A174725, A236914, A340101.

h := n -> add(combinat:-eulerian1(n,k)*2^k,k=0..n):
a := n -> (h(n)-(-1)^n)/2: seq(a(n),n=0..20); # Peter Luschny, Jul 09 2015

Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]

a(n)=if(n<0,0,n!*polcoeff(subst(y/(1-y^2),y,exp(x+x*O(x^n))-1),n))

{a(n)=polcoeff(sum(m=0,n,(2*m+1)!*x^(2*m+1)/prod(k=1,2*m+1,1-k*x+x*O(x^n))),n)} /* Paul D. Hanna, Jul 20 2011 */

def A089677_list(len):  # with a(0)=1
    e, r = [1], [1]
    for i in (1..len-1):
        for k in range(i-1, -1, -1): e[k] = (e[k]*i)//(i-k)
        r.append(-sum(e[j]*(-1)^(i-j) for j in (0..i-1)))
        e.append(sum(e))
    return r
A089677_list(21) # Peter Luschny, Jul 09 2015

E.g.f.: (exp(x)-1)/(exp(x)*(2-exp(x))).
O.g.f.: Sum_{n>=0} (2*n+1)! * x^(2*n+1) / Product_{k=1..2*n+1} (1-k*x). - Paul D. Hanna, Jul 20 2011
a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n).
a(n) = (A000670(n)-(-1)^n)/2. - Vladeta Jovovic, Jan 17 2005
a(n) ~ n! / (4*(log(2))^(n+1)). - Vaclav Kotesovec, Feb 25 2014
a(n) = Sum_{k=0..floor(n/2)} (2*k+1)!*Stirling2(n, 2*k+1). - Peter Luschny, Sep 20 2015

A342343 Number of strict compositions of n with alternating parts strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 10, 13, 18, 27, 32, 44, 55, 73, 97, 121, 151, 194, 240, 299, 384, 465, 576, 706, 869, 1051, 1293, 1572, 1896, 2290, 2761, 3302, 3973, 4732, 5645, 6759, 7995, 9477, 11218, 13258, 15597, 18393, 21565, 25319, 29703, 34701, 40478, 47278, 54985

Offset: 0

Gus Wiseman, Apr 01 2021

nonn

These are finite odd-length sequences q of distinct positive integers summing to n such that q(i) > q(i+2) for all possible i.

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
            (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
            (2,1)  (3,1)  (2,3)  (2,4)    (2,5)    (2,6)
                          (3,2)  (4,2)    (3,4)    (3,5)
                          (4,1)  (5,1)    (4,3)    (5,3)
                                 (2,3,1)  (5,2)    (6,2)
                                 (3,1,2)  (6,1)    (7,1)
                                 (3,2,1)  (2,4,1)  (2,5,1)
                                          (4,1,2)  (3,4,1)
                                          (4,2,1)  (4,1,3)
                                                   (4,3,1)
                                                   (5,1,2)
                                                   (5,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The non-strict case is A000041 (see A342528 for a bijective proof).
The non-strict odd-length case is A001522.
Strict compositions in general are counted by A032020
The non-strict even-length case is A064428.
The case of reversed partitions is A065033.
A000726 counts partitions with alternating parts unequal.
A003242 counts anti-run compositions.
A027193 counts odd-length compositions.
A034008 counts even-length compositions.
A064391 counts partitions by crank.
A064410 counts partitions of crank 0.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
A325548 counts compositions with strictly decreasing differences.
A342194 counts strict compositions with equal differences.
A342527 counts compositions with alternating parts equal.
Cf. A000009, A000670, A008965, A062968, A065608, A109613, A114921, A325546, A332304, A332305, A342532.

ici[q_]:=And@@Table[q[[i]]>q[[i+2]],{i,Length[q]-2}];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&],ici]],{n,0,15}]

seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=0, n, binomial(k, k\2) * polcoef(p,k,y)))} \\ Andrew Howroyd, Apr 16 2021

G.f.: Sum_{k>=0} binomial(k,floor(k/2)) * [y^k](Product_{j>=1} 1 + y*x^j). - Andrew Howroyd, Apr 16 2021

A339431 Number of compositions (ordered partitions) of n into an odd number of distinct squares.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 0, 1, 0, 0, 0, 0, 6, 0, 0, 0, 1, 6, 0, 0, 6, 6, 0, 0, 0, 0, 6, 1, 0, 6, 0, 0, 6, 6, 0, 0, 6, 6, 0, 0, 7, 6, 0, 0, 6, 6, 120, 6, 0, 0, 6, 0, 6, 12, 0, 1, 6, 126, 0, 0, 12, 6, 0, 0, 0, 12, 126, 0, 12, 6, 120, 0, 7

Offset: 0

Ilya Gutkovskiy, Dec 04 2020

			a(55) = 120 because we have [25, 16, 9, 4, 1] (120 permutations).

Cf. A000290, A331844, A332304, A339367, A339419, A339430.

b:= proc(n, i, p) option remember; `if`(n=0, irem(p, 2)*p!,
     (s-> `if`(s>n, 0, b(n, i+1, p)+b(n-s, i+1, p+1)))(i^2))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 04 2020

b[n_, i_, p_] := b[n, i, p] = If[n == 0, Mod[p, 2]*p!, With[{s = i^2}, If[s > n, 0, b[n, i + 1, p] + b[n - s, i + 1, p + 1]]]];
a[n_] := b[n, 1, 0];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)

A339433 Number of compositions (ordered partitions) of n into an odd number of distinct primes.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 6, 1, 6, 1, 6, 6, 6, 1, 12, 7, 12, 12, 12, 13, 12, 12, 18, 18, 132, 19, 132, 25, 18, 24, 252, 30, 258, 31, 264, 156, 486, 37, 498, 157, 504, 276, 738, 175, 738, 288, 750, 528, 984, 307, 1218, 666, 1110, 780, 6378, 679, 6618, 799, 1716, 1272

Offset: 0

Ilya Gutkovskiy, Dec 04 2020

nonn

			a(10) = 6 because we have [5, 3, 2], [5, 2, 3], [3, 5, 2], [3, 2, 5], [2, 5, 3] and [2, 3, 5].

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

Cf. A000040, A184172, A219107, A332304, A339383, A339409, A339432.

b:= proc(n, i, p) option remember; `if`(n=0, irem(p, 2)*p!, (s->
     `if`(s>n, 0, b(n, i+1, p)+b(n-s, i+1, p+1)))(ithprime(i)))
    end:
a:= n-> b(n, 1, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Dec 04 2020

b[n_, i_, p_] := b[n, i, p] = If[n == 0, Mod[p, 2]*p!, Function[s, If[s > n, 0, b[n, i + 1, p] + b[n - s, i + 1, p + 1]]][Prime[i]]];
a[n_] := b[n, 1, 0];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Feb 26 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A027193 Number of partitions of n into an odd number of parts.

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A067659 Number of partitions of n into distinct parts such that number of parts is odd.

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A174726 a(n) = (A002033(n-1) - A008683(n))/2.

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A340855 Numbers that can be factored into factors > 1, the least of which is odd.

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A340385 Number of integer partitions of n into an odd number of parts, the greatest of which is odd.

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A332305 Number of compositions (ordered partitions) of n into distinct parts such that number of parts is even.

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A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.

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