A078408 -id:A078408 - cp's OEIS frontend

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

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}]

A279374 Number of ways to choose an odd partition of each part of an odd partition of 2n+1.

Original entry on oeis.org

1, 3, 6, 15, 37, 80, 183, 428, 893, 1944, 4223, 8691, 18128, 37529, 75738, 153460, 308829, 612006, 1211097, 2386016, 4648229, 9042678, 17528035, 33645928, 64508161, 123178953, 233709589, 442583046, 834923483, 1567271495, 2935406996, 5481361193, 10191781534

Offset: 0

Gus Wiseman, Dec 11 2016

nonn

An odd partition is an integer partition of an odd number with an odd number of parts, all of which are odd.

			The a(3)=15 ways to choose an odd partition of each part of an odd partition of 7 are:
((7)), ((511)), ((331)), ((31111)), ((1111111)), ((5)(1)(1)), ((311)(1)(1)),
((11111)(1)(1)), ((3)(3)(1)), ((3)(111)(1)), ((111)(3)(1)), ((111)(111)(1)),
((3)(1)(1)(1)(1)), ((111)(1)(1)(1)(1)), ((1)(1)(1)(1)(1)(1)(1)).

Alois P. Heinz, Table of n, a(n) for n = 0..4919
Gus Wiseman, "Twice-odd partitions of n=9."

Cf. A000009 (strict partitions), A078408 (odd partitions), A063834, A271619, A279375.

g:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      [0, 2, 0, 1$4, 2, 0, 2, 1$4, 0, 2][1+irem(d, 16)],
      d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i, t) option remember;
      `if`(n=0, t, `if`(i<1, 0, b(n, i-2, t)+
      `if`(i>n, 0, b(n-i, i, 1-t)*g((i-1)/2))))
    end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 12 2016

nn=20;Table[SeriesCoefficient[Product[1/(1-PartitionsQ[k]x^k),{k,1,2n-1,2}],{x,0,2n-1}],{n,nn}]

A300300 Number of ways to choose a multiset of strict partitions, or odd partitions, of odd numbers, whose weights sum to n.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 9, 14, 20, 32, 48, 69, 105, 150, 225, 322, 472, 669, 977, 1379, 1980, 2802, 3977, 5602, 7892, 11083, 15494, 21688, 30147, 42007, 58143, 80665, 111199, 153640, 211080, 290408, 397817, 545171, 744645, 1016826, 1385124, 1885022, 2561111, 3474730

Offset: 0

Gus Wiseman, Mar 02 2018

nonn

			The a(6) = 9 multiset partitions using odd-weight strict partitions: (5)(1), (14)(1), (3)(3), (32)(1), (3)(21), (3)(1)(1)(1), (21)(21), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1).
The a(6) = 9 multiset partitions using odd partitions: (5)(1), (3)(3), (311)(1), (3)(111), (3)(1)(1)(1), (11111)(1), (111)(111), (111)(1)(1)(1), (1)(1)(1)(1)(1)(1).

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000009, A063834, A078408, A089259, A270995, A271619, A279374, A279375, A279785, A279790, A294617, A300301.

with(numtheory):
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
      `if`(d::odd, d, 0), d=divisors(j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(d::odd, b(d)*d, 0), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Mar 02 2018

nn=50;
ser=Product[1/(1-x^n)^PartitionsQ[n],{n,1,nn,2}];
Table[SeriesCoefficient[ser,{x,0,n}],{n,0,nn}]

Euler transform of {Q(1), 0, Q(3), 0, Q(5), 0, ...} where Q = A000009.

A300439 Number of odd enriched p-trees of weight n (all outdegrees are odd).

Original entry on oeis.org

1, 1, 2, 2, 5, 7, 18, 29, 75, 132, 332, 651, 1580, 3268, 7961, 16966, 40709, 89851, 215461, 484064, 1159568, 2641812, 6337448, 14622880, 35051341, 81609747, 196326305, 459909847, 1107083238, 2611592457, 6299122736, 14926657167, 36069213786, 85809507332

Offset: 1

Gus Wiseman, Mar 05 2018

nonn

An odd enriched p-tree of weight n > 0 is either a single node of weight n, or a finite odd-length sequence of at least 3 odd enriched p-trees whose weights are weakly decreasing and sum to n.

			The a(6) = 7 odd enriched p-trees: 6, (411), (321), (222), ((111)21), ((211)11), (21111).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000009, A027193, A063834, A078408, A196545, A273873, A289501, A294079, A298118, A299202, A299203, A300300, A300301, A300436, A300440.

f[n_]:=f[n]=1+Sum[Times@@f/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Array[f,40]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)) - 1/prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)/2); v} \\ Andrew Howroyd, Aug 26 2018

A358824 Number of twice-partitions of n of odd length.

Original entry on oeis.org

0, 1, 2, 4, 7, 15, 32, 61, 121, 260, 498, 967, 1890, 3603, 6839, 12972, 23883, 44636, 82705, 150904, 275635, 501737, 905498, 1628293, 2922580, 5224991, 9296414, 16482995, 29125140, 51287098, 90171414, 157704275, 275419984, 479683837, 833154673, 1442550486, 2493570655

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)        (4)         (5)
       (11)  (21)       (22)        (32)
             (111)      (31)        (41)
             (1)(1)(1)  (211)       (221)
                        (1111)      (311)
                        (2)(1)(1)   (2111)
                        (11)(1)(1)  (11111)
                                    (2)(2)(1)
                                    (3)(1)(1)
                                    (11)(2)(1)
                                    (2)(11)(1)
                                    (21)(1)(1)
                                    (11)(11)(1)
                                    (111)(1)(1)
                                    (1)(1)(1)(1)(1)

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

The version for set partitions is A024429.
For odd lengths (instead of length) we have A358334.
The case of odd parts also is A358823.
The case of odd sums also is A358826.
The case of odd lengths also is A358834.
For multiset partitions of integer partitions: A358837, ranked by A026424.
A000009 counts partitions into odd parts.
A027193 counts partitions of odd length.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
A078408 counts odd-length partitions into odd parts.
A300301 aerated counts twice-partitions with odd sums and parts.
Cf. A000041, A001970, A072233, A270995, A271619, A279374, A279785, A306319, A336342, A356932.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&]],{n,0,10}]

R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=vector(n,k,numbpart(k))); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022

G.f.: ((1/Product_{k>=1} (1-A000041(k)*x^k)) - (1/Product_{k>=1} (1+A000041(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Terms a(26) and beyond from Andrew Howroyd, Dec 30 2022

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

A300436 Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 12, 13, 35, 37, 98, 107, 304, 336, 927, 1037, 3010, 3367, 9585, 10924, 32126, 36438, 105589, 121045, 359691, 412789, 1211214, 1398168, 4188930, 4831708, 14315544, 16636297, 50079792, 58084208, 173370663, 202101971, 611487744, 712709423

Offset: 1

Gus Wiseman, Mar 05 2018

nonn

An odd p-tree of weight n > 0 is either a single node (if n = 1) or a finite sequence of at least 3 odd p-trees whose weights are weakly decreasing odd numbers summing to n.

			The a(7) = 5 odd p-trees: ((ooo)(ooo)o), (((ooo)oo)oo), ((ooooo)oo), ((ooo)oooo), (ooooooo).

Cf. A000009, A027193, A063834, A078408, A196545, A279374, A279785, A289501, A298118, A299202, A299203, A300300, A300301, A300355, A300439, A300440.

b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&And@@OddQ/@#&]}];
Table[b[n],{n,40}]

O.g.f: x + Product_{n odd} 1/(1 - a(n)*x^n) - Sum_{n odd} a(n)*x^n. - Gus Wiseman, Aug 27 2018

Name corrected by Gus Wiseman, Aug 27 2018

A358334 Number of twice-partitions of n into odd-length partitions.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 43, 77, 137, 241, 410, 720, 1209, 2073, 3498, 5883, 9768, 16413, 26978, 44741, 73460, 120462, 196066, 320389, 518118, 839325, 1353283, 2178764, 3490105, 5597982, 8922963, 14228404, 22609823, 35875313, 56756240, 89761600, 141410896, 222675765

Offset: 0

Gus Wiseman, Dec 01 2022

nonn

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(0) = 1 through a(5) = 13 twice-partitions:
  ()  ((1))  ((2))     ((3))        ((4))           ((5))
             ((1)(1))  ((111))      ((211))         ((221))
                       ((2)(1))     ((2)(2))        ((311))
                       ((1)(1)(1))  ((3)(1))        ((3)(2))
                                    ((111)(1))      ((4)(1))
                                    ((2)(1)(1))     ((11111))
                                    ((1)(1)(1)(1))  ((111)(2))
                                                    ((211)(1))
                                                    ((2)(2)(1))
                                                    ((3)(1)(1))
                                                    ((111)(1)(1))
                                                    ((2)(1)(1)(1))
                                                    ((1)(1)(1)(1)(1))

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

For multiset partitions of integer partitions: A356932, ranked by A356935.
For odd length instead of lengths we have A358824.
For odd sums instead of lengths we have A358825.
For odd sums also we have A358827.
For odd length also we have A358834.
A000041 counts integer partitions.
A027193 counts odd-length partitions, ranked by A026424.
A055922 counts partitions with odd multiplicities, also odd parts A117958.
A063834 counts twice-partitions, strict A296122, row-sums of A321449.
Cf. A000219, A001970, A072233, A078408, A270995, A279374, A298118, A300300, A300301, A300647, A302243.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&]],{n,0,10}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1), -(n+1))} \\ Andrew Howroyd, Dec 30 2022

G.f.: 1/Product_{k>=1} (1 - A027193(k)*x^k). - Andrew Howroyd, Dec 30 2022

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

A152157 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of partitions of 2n+1 into 2k+1 odd parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 3, 2, 1, 1, 1, 5, 5, 3, 2, 1, 1, 1, 7, 7, 5, 3, 2, 1, 1, 1, 8, 10, 7, 5, 3, 2, 1, 1, 1, 10, 13, 11, 7, 5, 3, 2, 1, 1, 1, 12, 18, 15, 11, 7, 5, 3, 2, 1, 1, 1, 14, 23, 21, 15, 11, 7, 5, 3, 2, 1, 1, 1, 16, 30, 28, 22, 15, 11, 7, 5, 3, 2, 1, 1

Offset: 0

R. J. Mathar, Sep 25 2009

This number triangle satisfies T(n,k) = A008284(n+k+1,2*k+1), n,k >= 0. This means that T(n,k) is also the number of partitions of N:=n+k+1 into M:=2*k+1 parts. For the proof add 1 to every odd part of each partition of n':=2*n+1 into m:=2*k+1 parts which are all odd, and divide each part by a factor of 2, thus obtaining a partition of n+k+1 into m=2*k+1 parts. All partitions of N,for N>=1, into an odd number of parts M (M from {1,...,N}) are reached: just take k=(M-1)/2 and n=N-1-k. Each partition of N into an odd number of parts can only arise once from the given recipe (for given N and M the k and n values are unique). See also a comment by Franklin T. Adams-Watters on A152140. - Wolfdieter Lang, Jul 09 2012

			Triangle begins:
1
1  1
1  1   1
1  2   1   1
1  3   2   1   1
1  4   3   2   1   1
1  5   5   3   2   1   1
1  7   7   5   3   2   1  1
1  8  10   7   5   3   2  1  1
1 10  13  11   7   5   3  2  1  1
1 12  18  15  11   7   5  3  2  1  1
1 14  23  21  15  11   7  5  3  2  1  1
1 16  30  28  22  15  11  7  5  3  2  1  1
1 19  37  38  30  22  15 11  7  5  3  2  1  1
1 21  47  49  41  30  22 15 11  7  5  3  2  1 1
1 24  57  65  54  42  30 22 15 11  7  5  3  2 1 1
1 27  70  82  73  56  42 30 22 15 11  7  5  3 2 1  1
1 30  84 105  94  76  56 42 30 22 15 11  7  5 3 2  1 1
1 33 101 131 123  99  77 56 42 30 22 15 11  7 5 3  2 1 1
1 37 119 164 157 131 101 77 56 42 30 22 15 11 7 5  3 2 1 1
From _Wolfdieter Lang_, Jul 09 2012 (Start)
T(5,1) = 4 from the four partitions of 11 into 3 parts, all of which are odd: [1,1,9], [1,3,7], [1,5,5] and [3,3,5].
T(5,1) = 4 from the four partitions of 7 = 5+1+1 into 3 parts:
[1,1,5], [1,2,4], [1,3,3] and [2,2,3].
(End)

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A078408 (row sums), A107379, A152140, A152146, A008284.

b:= proc(n, i) option remember; `if`(n=0, 1/sqrt(x), `if`(i<1, 0,
      b(n, i-2)+`if`(i>n, 0, expand(sqrt(x)*b(n-i, i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2*n+1, 2*n+1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jun 21 2021

(* p = A008284 *) p[n_, 1] = 1; p[n_, k_] := p[n, k] = If[n >= k, Sum[p[n - i, k - 1], {i, 1, n - 1}] - Sum[p[n - i, k], {i, 1, k - 1}], 0];
T[n_, k_] := p[n + k + 1, 2 k + 1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 28 2019, after Wolfdieter Lang *)

T(n,k) = A152140(2n+1,2k+1).
T(n,k) = p(n+k+1,2*k+1), n >= 0, k >= 0, with p(N,M)= A008284(N,M), the number of partitions of N into M parts. See the sketch of the proof given above as a comment. - Wolfdieter Lang, Jul 09 2012
O.g.f. for column k: (x^k)/product(1-x^j,j=1..(2*k+1)), k>=0.
  From the o.g.f.s of A008284. - Wolfdieter Lang, Jul 10 2012

Indices corrected by R. J. Mathar, Jul 09 2012

A300574 Coefficient of x^n in 1/((1-x)(1+x^3)(1-x^5)(1+x^7)(1-x^9)...).

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 2, 1, 0, 0, 2, 2, 1, 0, 2, 3, 2, 0, 2, 4, 4, 0, 1, 4, 6, 2, 1, 4, 8, 4, 2, 4, 10, 6, 2, 3, 12, 10, 4, 2, 13, 14, 8, 2, 14, 18, 12, 2, 14, 22, 18, 3, 14, 26, 26, 6, 14, 29, 34, 10, 14, 32, 44, 16, 14, 34, 56, 26, 16, 34, 67, 38, 20, 34, 78, 52, 26

Offset: 0

Gus Wiseman, Mar 08 2018

By Theorem 1 of Craig, the values a(n) in this list are known to be nonnegative. Combined with Theorem 2 of Seo and Yee, this shows that a(n) = |number of partitions of n into odd parts with an odd index minus the number of partitions of n into odd parts with an even index|. - William Craig, Dec 31 2021

Seunghyun Seo and Ae Ja Yee, Index of seaweed algebras and integer partitions, Electronic Journal of Combinatorics, 27:1 (2020), #P1.47. See Conjecture 1 and Theorem 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Gus Wiseman)
Vincent E. Coll, Andrew W. Mayers, and Nick W. Mayers, Statistics on integer partitions arising from seaweed algebras, arXiv:1809.09271 [math.CO], 2018.
William Craig, Seaweed Algebras and the Index Statistic for Partitions, arXiv:2112.09269 [math.CO], 2021.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000009, A010815, A027193, A067659, A078408, A081362, A099323, A220418, A290261, A292043, A292137, A298118, A300575.

CoefficientList[Series[1/QPochhammer[x, -x^2], {x, 0, 100}], x]
nmax = 100; CoefficientList[Series[Product[1/((1+x^(4*k-1))*(1-x^(4*k-3))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 04 2019 *)

O.g.f.: Product_{n >= 0} 1/(1 - (-1)^n x^(2n+1)).
a(n) = Sum (-1)^k where the sum is over all integer partitions of n into odd parts and k is the number of parts not congruent to 1 modulo 4.
a(n) has average order Gamma(1/4) * exp(sqrt(n/3)*Pi/2) / (2^(9/4) * 3^(1/8) * Pi^(3/4) * n^(5/8)). - Vaclav Kotesovec, Jun 04 2019

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