A298118 -id:A298118 - cp's OEIS frontend

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

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

A301344 Regular triangle where T(n,k) is the number of semi-binary rooted trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 1, 0, 0, 1, 6, 4, 0, 0, 0, 1, 9, 11, 2, 0, 0, 0, 1, 12, 24, 9, 0, 0, 0, 0, 1, 16, 46, 32, 3, 0, 0, 0, 0, 1, 20, 80, 86, 20, 0, 0, 0, 0, 0, 1, 25, 130, 203, 86, 6, 0, 0, 0, 0, 0, 1, 30, 200, 423, 283, 46, 0, 0, 0, 0, 0, 0, 1, 36, 295, 816, 786, 234, 11, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Mar 19 2018

A rooted tree is semi-binary if all outdegrees are <= 2. The number of semi-binary trees with n nodes is equal to the number of binary trees with n+1 leaves; see A001190.

			Triangle begins:
1
1   0
1   1   0
1   2   0   0
1   4   1   0   0
1   6   4   0   0   0
1   9  11   2   0   0   0
1  12  24   9   0   0   0   0
1  16  46  32   3   0   0   0   0
1  20  80  86  20   0   0   0   0   0
1  25 130 203  86   6   0   0   0   0   0
The T(6,3) = 4 semi-binary rooted trees: ((o(oo))), (o((oo))), (o(o(o))), ((o)(oo)).

Cf. A000081, A000598, A001190, A001678, A003238, A004111, A055277, A111299, A273873, A290689, A291636, A292050, A298118, A298204, A298422, A298426, A301342, A301343, A301345.

rbt[n_]:=rbt[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rbt/@c]]]/@Select[IntegerPartitions[n-1],Length[#]<=2&]];
Table[Length[Select[rbt[n],Count[#,{},{-2}]===k&]],{n,15},{k,n}]

A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

Original entry on oeis.org

0, 1, 1, 0, 0, 0, -1, -2, -2, -2, -2, -3, -3, -2, -2, -2, -1, 1, 2, 2, 4, 6, 7, 8, 10, 13, 14, 14, 15, 17, 17, 15, 15, 16, 14, 10, 8, 6, 1, -5, -10, -14, -21, -31, -38, -43, -53, -64, -71, -77, -86, -97, -104, -108, -115, -124, -127, -125, -127, -130, -125, -116

Offset: 0

Seiichi Manyama, Sep 09 2017

			Product_{k>=1} 1/(1 - i*x^k) = 1 + (0+1i)*x + (-1+1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2-1i)*x^6 + (-1-2i)*x^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A000108, A010815, A027193, A063834, A067659, A081362, A099323, A196545, A220418, A290261, A292042, A292043, A292136, A292138, A298118, A300355.

N:= 100:
S := convert(series( add( (-1)^n*x^(2*n+1)/(mul(1 - x^k,k = 1..2*n+1)), n = 0..N ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021

Im[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 17 2017 *)

1/(i*x; x)_inf is the g.f. for A292136(n) + i*a(n).
a(n) = Sum (-1)^((k - 1)/2) where the sum is over all integer partitions of n into an odd number of parts and k is the number of parts. - Gus Wiseman, Mar 08 2018
G.f.: Sum_{n >= 0} (-1)^n * x^(2*n+1)/Product_{k = 1..2*n+1} (1 - x^k). - Peter Bala, Jan 15 2021

A298304 Number of rooted trees on n nodes with strictly thinning limbs.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 7, 12, 19, 31, 51, 85, 144, 245, 417, 712, 1221, 2091, 3600, 6216, 10763, 18691, 32546, 56782, 99271, 173849, 304877, 535412, 941385, 1657069, 2919930, 5150546, 9093894, 16071634, 28428838, 50331137, 89181251, 158145233, 280650225, 498410197

Offset: 1

Gus Wiseman, Jan 16 2018

nonn

An unlabeled rooted tree has strictly thinning limbs if its outdegrees are strictly decreasing from root to leaves.

			The a(7) = 7 trees: (oo(o(o))), (o(o)(oo)), (ooo(oo)), ((o)(o)(o)), (oo(o)(o)), (oooo(o)), (oooooo).

Alois P. Heinz, Table of n, a(n) for n = 1..300

Cf. A000081, A001678, A004111, A032305, A124343, A290689, A295461, A298118, A298303, A298305.

stinctQ[t_]:=And@@Cases[t,b_List:>Length[b]>Max@@Length/@b,{0,Infinity}];
strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],stinctQ]];
Table[Length[strut[n]],{n,20}]

a(26)-a(40) from Alois P. Heinz, Jan 17 2018

A300647 Number of same-trees of weight n in which all outdegrees are odd.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 10, 2, 2, 2, 2, 2, 42, 1, 2, 10, 2, 2, 138, 2, 2, 2, 34, 2, 1514, 2, 2, 42, 2, 1, 2058, 2, 162, 10, 2, 2, 8202, 2, 2, 138, 2, 2, 207370, 2, 2, 2, 130, 34, 131082, 2, 2, 1514, 2082, 2, 524298, 2, 2, 42, 2, 2, 14725738, 1, 8226, 2058, 2

Offset: 1

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(9) = 10 odd same-trees:
9,
(333),
(33(111)), (3(111)3), ((111)33)
(3(111)(111)), ((111)3(111)), ((111)(111)3),
((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, A300648, A300649, A300650.

a[n_]:=1+Sum[a[n/d]^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), a(n/d)^d))); \\ Michel Marcus, Mar 10 2018

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

A300862 Solution to 1 = Sum_y Product_{k in y} a(k) for each n > 0, where the sum is over all integer partitions of n with an odd number of parts.

Original entry on oeis.org

1, 1, 0, 0, -1, -1, 0, 1, 1, 0, -2, -3, -2, 2, 7, 6, -3, -15, -19, -2, 32, 54, 24, -64, -153, -123, 95, 389, 444, -43, -966, -1475, -516, 2066, 4414, 3092, -3874, -12480, -12936, 3847, 32445, 45494, 8950, -77282, -147663, -86313, 157456, 435623, 399041, -229616, -1211479, -1535700, -73132

Offset: 1

Gus Wiseman, Mar 13 2018

sign

Cf. A027193, A063834, A220418, A279374, A290261, A290971, A298118, A299202, A299203, A300301, A300436, A300439, A300863, A300864, A300865, A300866.

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

A300440 Number of odd strict trees of weight n (all outdegrees are odd).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 5, 7, 11, 18, 27, 45, 75, 125, 207, 353, 591, 1013, 1731, 2984, 5122, 8905, 15369, 26839, 46732, 81850, 142932, 251693, 441062, 778730, 1370591, 2425823, 4281620, 7601359, 13447298, 23919512, 42444497, 75632126, 134454505, 240100289

Offset: 1

Gus Wiseman, Mar 05 2018

nonn

An odd strict tree of weight n is either a single node of weight n, or a finite odd-length sequence of at least 3 odd strict trees with strictly decreasing weights summing to n.

			The a(10) = 7 odd strict trees: 10, (721), (631), (541), (532), ((421)21), ((321)31).

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

Cf. A000009, A000992, A032305, A063834, A078408, A089259, A196545, A273873, A279785, A289501, A298118, A300301, A300352, A300353, A300436, A300439.

g[n_]:=g[n]=1+Sum[Times@@g/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&&UnsameQ@@#&]}];
Array[g,20]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)) - prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)/2); v} \\ Andrew Howroyd, Aug 25 2018

A298126 Matula-Goebel numbers of rooted trees in which all outdegrees are even.

Original entry on oeis.org

1, 4, 14, 16, 49, 56, 64, 86, 106, 196, 224, 256, 301, 344, 371, 424, 454, 526, 622, 686, 784, 886, 896, 1024, 1154, 1204, 1376, 1484, 1589, 1696, 1816, 1841, 1849, 2104, 2177, 2279, 2386, 2401, 2488, 2744, 2809, 2846, 3101, 3136, 3238, 3544, 3584, 3986, 4039

Offset: 1

Gus Wiseman, Jan 13 2018

nonn

			Sequence of trees begins:
1   o
4   (oo)
14  (o(oo))
16  (oooo)
49  ((oo)(oo))
56  (ooo(oo))
64  (oooooo)
86  (o(o(oo)))
106 (o(oooo))
196 (oo(oo)(oo))
224 (ooooo(oo))
256 (oooooooo)
301 ((oo)(o(oo)))
344 (ooo(o(oo)))
371 ((oo)(oooo))
424 (ooo(oooo))
454 (o((oo)(oo)))
526 (o(ooo(oo)))
622 (o(oooooo))
686 (o(oo)(oo)(oo))
784 (oooo(oo)(oo))
886 (o(o(o(oo))))
896 (ooooooo(oo))

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A001190, A007097, A026424, A028260, A061775, A111299, A214577, A245824, A276625, A277098, A290760, A291441, A291442, A291636, A295461, A297571, A298118, A298120.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
etQ[n_]:=Or[n===1,With[{m=primeMS[n]},EvenQ@Length@m&&And@@etQ/@m]];
Select[Range[10000],etQ]

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.

A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.

Original entry on oeis.org

1, 1, 2, 6, 12, 30, 82, 198, 533, 1459, 4039, 11634, 34095, 102520, 316456, 995709, 3215552, 10591412, 35633438, 122499429, 428988392, 1532929060, 5579867442, 20677066725, 78027003260, 299413756170, 1168536196157, 4635420192861, 18678567555721, 76451691937279, 317625507668759

Offset: 0

Gus Wiseman, Oct 18 2018

nonn

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears an odd number of times.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions with all parts of odd size:
  {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
         {{1},{2}}  {{1,2,2}}      {{1},{1,2,2}}
                    {{1,2,3}}      {{1},{2,2,2}}
                    {{1},{1},{1}}  {{1},{2,3,3}}
                    {{1},{2},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A001055, A001222, A007716, A298118, A300300, A300301, A318871, A320663, A320665.

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp((A-subst(A,x,-x))/2)))} \\ Andrew Howroyd, Jan 17 2023

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
J(q, t, k, y)={1/prod(j=1, #q, my(s=q[j], g=gcd(s,t)); (1 + O(x*x^k) - y^(s/g)*x^(s*t/g))^g)}
K(q, t, k) = Vec(J(q,t,k,1)-J(q,t,k,-1), -k)/2
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 17 2023

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

cp's OEIS Frontend

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

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A301344 Regular triangle where T(n,k) is the number of semi-binary rooted trees with n nodes and k leaves.

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Mathematica

A292137 G.f.: Im(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

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Maple

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A298304 Number of rooted trees on n nodes with strictly thinning limbs.

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Mathematica

Extensions

A300647 Number of same-trees of weight n in which all outdegrees are odd.

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PARI

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A300862 Solution to 1 = Sum_y Product_{k in y} a(k) for each n > 0, where the sum is over all integer partitions of n with an odd number of parts.

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Mathematica

A300440 Number of odd strict trees of weight n (all outdegrees are odd).

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PARI

A298126 Matula-Goebel numbers of rooted trees in which all outdegrees are even.

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