A300647 -id:A300647 - cp's OEIS frontend

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

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

A300648 Number of orderless same-trees of weight n in which all outdegrees are odd.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 12, 1, 2, 6, 2, 2, 14, 2, 2, 2, 8, 2, 68, 2, 2, 12, 2, 1, 18, 2, 16, 6, 2, 2, 20, 2, 2, 14, 2, 2, 644, 2, 2, 2, 10, 8, 24, 2, 2, 68, 20, 2, 26, 2, 2, 12, 2, 2, 1386, 1, 22, 18, 2, 2, 30, 16, 2, 6, 2, 2, 4532, 2, 22, 20

Offset: 1

Gus Wiseman, Mar 10 2018

An orderless same-tree of weight n > 0 is either a single node of weight n, or a finite multiset of two or more orderless same-trees whose weights are all equal and sum to n.

			The a(9) = 6 odd orderless same-trees: 9, (333), (33(111)), (3(111)(111)), ((111)(111)(111)), (111111111).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300647, A300649, A300650.

a[n_]:=1+Sum[Binomial[a[n/d]+d-1,d],{d,Select[Rest[Divisors[n]],OddQ]}];
Array[a,80]

a(n) = if (n==1, 1, 1 + sumdiv(n, d, if ((d > 1) && (d % 2), binomial(a(n/d) + d - 1, d)))); \\ Michel Marcus, Mar 10 2018

a(n) = 1 + Sum_d binomial(a(n/d) + d - 1, d) where the sum is over odd divisors of n greater than 1.

A300649 Number of same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 10, 1, 1, 3, 3, 1, 3, 1, 1, 62, 1, 2, 3, 1, 3, 3, 1, 1, 158, 3, 1, 3, 1, 1, 254, 3, 1, 1514, 1, 3, 3, 1, 3, 3, 3, 1, 2078, 1, 1, 2461, 1, 1, 3, 1, 3, 8222, 3, 2, 3, 34, 1, 3, 1, 3, 390782, 1, 1, 3, 3, 3, 2198, 1, 1

Offset: 0

Gus Wiseman, Mar 10 2018

nonn

A same-tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more same-trees whose weights are all equal and sum to n.

			The a(13) = 10 odd same-trees with all leaves greater than 1:
27,
(999),
(99(333)), (9(333)9), ((333)99),
(9(333)(333)), ((333)9(333)), ((333)(333)9),
((333)(333)(333)), (333333333).

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300647, A300648, A300650.

a[n_]:=If[n===1,1,Sum[a[n/d]^d,{d,Select[Rest[Divisors[n]],OddQ]}]];
Table[a[n],{n,1,100,2}]

f(n) = if (n==1, 1, sumdiv(n, d, if ((d > 1) && (d % 2), f(n/d)^d)));
a(n) = f(2*n+1); \\ Michel Marcus, Mar 10 2018

a(1) = 1; a(n > 1) = Sum_d a(n/d)^d where the sum is over odd divisors of n greater than 1.

A300650 Number of orderless same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 6, 1, 1, 3, 3, 1, 3, 1, 1, 19, 1, 2, 3, 1, 3, 3, 1, 1, 21, 3, 1, 3, 1, 1, 28, 3, 1, 68, 1, 3, 3, 1, 3, 3, 3, 1, 25, 1, 1, 71, 1, 1, 3, 1, 3, 27, 3, 2, 3, 8, 1, 3, 1, 3, 1656, 1, 1, 3, 3, 3, 43, 1, 1, 31, 3, 1, 3, 3, 1

Offset: 0

Gus Wiseman, Mar 10 2018

nonn

An orderless same-tree of weight n > 0 is either a single node of weight n, or a finite multiset of two or more orderless same-trees whose weights are all equal and sum to n.

			The a(13) = 6 orderless same-trees: 27, (999), (99(333)), (9(333)(333)), ((333)(333)(333)), (333333333).

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300647, A300648, A300649.

a[n_]:=If[n===1,1,Sum[Binomial[a[n/d]+d-1,d],{d,Select[Rest[Divisors[n]],OddQ]}]];
Table[a[n],{n,1,100,2}]

f(n) = if (n==1, 1, sumdiv(n, d, if ((d > 1) && (d % 2), binomial(f(n/d)+d-1, d))));
a(n) = f(2*n+1); \\ Michel Marcus, Mar 10 2018

a(1) = 1; a(n > 1) = Sum_d binomial(a(n/d) + d - 1, d) where the sum is over odd divisors of n greater than 1.

A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 107, 336, 1037, 3367, 10924, 36438, 121045, 412789, 1398168, 4831708, 16636297, 58084208, 202101971, 712709423, 2502000811, 8880033929, 31428410158, 112199775788, 399383181020, 1433385148187, 5128572792587, 18481258241133

Offset: 0

Gus Wiseman, Aug 27 2018

nonn

A p-tree of weight n with odd outdegrees is either a single node (if n = 1) or a finite odd-length sequence of at least 3 p-trees with odd outdegrees whose weights are weakly decreasing and sum to n.

			The a(4) = 13 p-trees of weight 9 with odd outdegrees:
  ((((ooo)oo)oo)oo)
  (((ooo)(ooo)o)oo)
  (((ooo)oo)(ooo)o)
  ((ooo)(ooo)(ooo))
  (((ooooo)oo)oo)
  (((ooo)oooo)oo)
  ((ooooo)(ooo)o)
  (((ooo)oo)oooo)
  ((ooo)(ooo)ooo)
  ((ooooooo)oo)
  ((ooooo)oooo)
  ((ooo)oooooo)
  (ooooooooo)

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A027193, A063834, A078408, A196545, A279374, A289501, A298118, A300300, A300301, A300355, A300436, A300647, A300652, A300797, A302243.

b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Table[b[n],{n,1,20,2}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^(2*k-1) + O(x^(2*n))) - 1/prod(k=1, n-1, 1 + v[k]*x^(2*k-1) + O(x^(2*n))), 2*n-1)/2); v} \\ Andrew Howroyd, Aug 27 2018

cp's OEIS Frontend

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

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A300648 Number of orderless same-trees of weight n in which all outdegrees are odd.

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A300649 Number of same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.

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A300650 Number of orderless same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.

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A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.

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