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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

Andrew Howroyd, <a href="/A358824/b358824.txt">Table of n, a(n) for n = 0..1000</a>

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

(PARI)

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

nonn,more

nonn

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

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The version for For multiset partitions of integer partitions is : A358837, ranked by A026424.

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The case of odd parts also is A358823.

The case of odd sums also is A358826.

The case of odd lengths also is A358834.

Cf. A000041, A001970, `A072233, A270995, A271619, A279374, A279785, `A306319, `A336342, A356932.

The version for set partitions is A024429.

allocated for Gus WisemanNumber of twice-partitions of n of odd length.

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

0,3

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)

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];

Table[Length[Select[twiptn[n], OddQ[Length[#]]&]], {n, 0, 10}]

For odd lengths (instead of length) we have A358334.

The case of odd parts is A358823.

The case of odd sums is A358826.

The case of odd lengths is A358834.

The version for multiset partitions of integer partitions is A358837.

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.

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Gus Wiseman, Dec 03 2022

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