A006241 -id:A006241 - cp's OEIS frontend

A281145 Number of same-trees of weight n.

1, 2, 2, 6, 2, 14, 2, 54, 10, 38, 2, 494, 2, 134, 42, 4470, 2, 3422, 2, 10262, 138, 2054, 2, 490926, 34, 8198, 1514, 314294, 2, 628318, 2, 30229110, 2058, 131078, 162, 150147342, 2, 524294, 8202, 628073814, 2, 109952254, 2, 371210294, 207370, 8388614, 2

Offset: 1

Gus Wiseman, Jan 15 2017

nonn

A same-tree is either: (case 1) a positive integer, or (case 2) a finite sequence of two or more same-trees all having the same weight, where the weight in case 2 is the sum of weights.

			The a(6)=14 same-trees are:
6,
(33),
(222),
(3(111)), ((111)3),
(22(11)), (2(11)2), ((11)22),
(2(11)(11)), ((11)2(11)), ((11)(11)2),
((111)(111)), ((11)(11)(11)), (111111).
The a(9)=10 same-trees are:
9,
(333),
(33(111)), (3(111)3), ((111)33),
(3(111)(111)), ((111)3(111)), ((111)(111)3),
((111)(111)(111)), (111111111).

G. C. Greubel, Table of n, a(n) for n = 1..4000

Cf. A196545, A260685, A273873, A275870, A006241.

a[n_]:=1+DivisorSum[n,b[#]^(n/#)&]-b[n]/.b->a;
Array[a,47]

seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sumdiv(n, d, v[n/d]^d)); v} \\ Andrew Howroyd, Aug 20 2018

a(n) = 1 + Sum a(d)^(n/d) where the sum is over divisors less than n.

A297791 Number of series-reduced leaf-balanced rooted trees with n nodes. Number of orderless same-trees with n nodes and all leaves equal to 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 3, 3, 4, 3, 5, 3, 6, 4, 6, 3, 12, 3, 10, 7, 9, 6, 12, 9, 13, 16, 14, 22, 22, 24, 21, 24, 28, 14, 32, 15, 42, 20, 60, 27, 84, 44, 100, 59, 113, 74, 116, 85, 110, 97, 96, 113, 106, 149, 147, 234, 235, 377, 380, 580, 576, 838

Offset: 1

Gus Wiseman, Jan 06 2018

nonn

An unlabeled rooted tree is leaf-balanced if all branches from the same root have the same number of leaves. It is series-reduced if all positive out-degrees are greater than one.

			The a(13) = 5 trees: (((oo)(oo))(oooo)), ((ooooo)(ooooo)), ((ooo)(ooo)(ooo)), ((oo)(oo)(oo)(oo)), (oooooooooooo).

Robert G. Wilson v, Table of n, a(n) for n = 1..84

Cf. A000081, A001190, A001678, A006241, A032305, A289078, A289079, A291441, A291443.

alltim[n_]:=alltim[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[alltim/@c]],And[SameQ@@(Count[#,{},{0,Infinity}]&/@#),FreeQ[#,{_}]]&]]/@IntegerPartitions[n-1]];
Table[Length[alltim[n]],{n,20}]

lista(nn) = my(k, r, t, u, w=vector(nn, i, vector(i))); w[1][1]=1; for(s=2, nn, fordiv(s, d, if(dw[i][d], [d..nn]); forvec(v=vector(s/d, i, [1, #u]), if(nn>=r=1+sum(i=1, #v, u[v[i]]), k=1; t=1; for(i=2, #v, if(v[i]==v[i-1], k++, t*=binomial(w[u[v[i-1]]][d]+k-1, k); k=1)); w[r][s]+=t*binomial(w[u[v[#v]]][d]+k-1, k)), 1)))); vector(nn, i, vecsum(w[i])); \\ Jinyuan Wang, Feb 25 2025

a(51) onward from Robert G. Wilson v, Jan 07 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.

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.

A294019 Number of same-trees whose leaves are the parts of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 3, 1, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 8

Offset: 1

Gus Wiseman, Feb 07 2018

nonn

By convention a(1) = 0.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(108) = 8 same-trees: ((22)(2(11))), ((22)((11)2)), ((2(11))(22)), (((11)2)(22)), (222(11)), (22(11)2), (2(11)22), ((11)222).
From _Antti Karttunen_, Sep 22 2018: (Start)
For 12 = prime(1)^2 * prime(2)^1, we have the following two cases: 2(11) and (11)2, thus a(12) = 2.
For 36 = prime(1)^2 * prime(2)^2, we have the following cases: (11)22, 2(11)2, 22(11), thus a(36) = 3.
For 144  = prime(1)^4 * prime(2)^2, we have the following 14 cases: (1111)(22), (22)(1111); ((11)(11))(22), (22)((11)(11)); (11)(11)22, (11)2(11)2, (11)22(11), 2(11)2(11), 2(11)(11)2, 22(11)(11); ((11)2)(11(2)), ((11)2)(2(11)), (2(11))((11)2), (2(11))(2(11)), thus a(144) = 14.
For n = 8775 = 3^3 * 5^2 * 13^1 = prime(2)^3 * prime(3)^2 * prime(6)^1, we have the following six cases: (222)(33)6, (222)6(33), (33)(222)6, (33)6(222), 6(222)(33), 6(33)(222), thus a(8775) = 6.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A000041, A000720, A001222, A006241, A056239, A063834, A196545, A215366, A273873, A281145, A289501, A296150, A299200, A299201, A299202, A299203, A294018, A294080.

nn=120;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
qci[y_]:=qci[y]=If[Length[y]===1,1,Sum[Times@@qci/@t,{t,Select[tris,And[Length[#]>1,Sort[Join@@#,Greater]===y,SameQ@@Total/@#]&]}]];
qci/@ptns

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
productifbalancedfactorization(v) = if(!#v, 1, my(pw=A056239(v[1]), m=1); for(i=1,#v,if(A056239(v[i])!=pw,return(0), m *= A294019(v[i]))); (m));
A294019aux(n, m, facs) = if(1==n, productifbalancedfactorization(Vec(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A294019aux(n/d, m, newfacs))); (s));
A294019(n) = if(1==n,0,if(isprime(n),1,A294019aux(n, n-1, List([]))));
\\ A memoized implementation:
map294019 = Map();
A294019(n) = if(1==n,0,if(isprime(n),1,if(mapisdefined(map294019,n), mapget(map294019,n), my(v=A294019aux(n, n-1, List([]))); mapput(map294019,n,v); (v)))); \\ Antti Karttunen, Sep 22 2018

A281145(n) = Sum_{i=1..A000041(n)} a(A215366(n,i)).

a(p^n) = A006241(n) for any prime p and exponent n >= 1. - Antti Karttunen, Sep 22 2018

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.

A301367 Regular triangle where T(n,k) is the number of orderless same-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 3, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 4, 4, 3, 5, 1, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 0, 1, 2, 1, 1, 1, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 4, 5, 10, 11, 14, 12, 14, 7, 13, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1

Offset: 1

Gus Wiseman, Mar 19 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 the same and sum to n.

			Triangle begins:
1
1   1
1   0   1
1   1   1   2
1   0   0   0   1
1   1   1   2   1   3
1   0   0   0   0   0   1
1   1   1   3   4   4   3   5
1   0   1   0   1   0   1   0   2
1   1   0   0   1   2   1   1   1   3
1   0   0   0   0   0   0   0   0   0   1
1   1   2   4   5  10  11  14  12  14   7  13
1   0   0   0   0   0   0   0   0   0   0   0   1
1   1   0   0   0   0   1   2   1   1   1   1   1   3
The T(8,5) = 4 orderless same-trees: (4((11)(11))), (4(1111)), ((22)(2(11))), (222(11)).

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Last entries of each row give A289079. Row sums are A289078.

Cf. A003238, A006241, A008284, A055277, A063834, A273873, A281145, A289501, A294080, A298422, A298426, A299201, A299203, A301343, A301364-A301368.

olstrees[n_]:=Prepend[Join@@Table[Select[Tuples[olstrees/@ptn],OrderedQ],{ptn,Select[IntegerPartitions[n],Length[#]>1&&SameQ@@#&]}],n];
Table[Length[Select[olstrees[n],Count[#,_Integer,{-1}]===k&]],{n,14},{k,n}]

S(g, k)={polcoef(exp(sum(i=1, k, x^i*subst(g, y, y^i)/i) + O(x*x^k)), k)}
A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + sumdiv(n, d, S(v[n/d], d))); apply(p -> Vecrev(p/y), v)}
{ my(v=A(16)); for(n=1, #v, print(v[n])) } \\ Andrew Howroyd, Aug 20 2018

A301366 Regular triangle where T(n,k) is the number of same-trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 1, 1, 5, 3, 3, 1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 6, 12, 14, 12, 6, 1, 0, 1, 0, 3, 0, 3, 0, 2, 1, 1, 0, 0, 1, 7, 10, 10, 5, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 7, 21, 41, 58, 100, 100, 94, 48, 20

Offset: 1

Gus Wiseman, Mar 19 2018

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 the same and sum to n.

			Triangle begins:
1
1   1
1   0   1
1   1   2   2
1   0   0   0   1
1   1   1   5   3   3
1   0   0   0   0   0   1
1   1   2   6  12  14  12   6
1   0   1   0   3   0   3   0   2
1   1   0   0   1   7  10  10   5   3
1   0   0   0   0   0   0   0   0   0   1
1   1   3   7  21  41  58 100 100  94  48  20
The T(8,4) = 6 same-trees: (4(2(11))), (4((11)2)), ((22)(22)), ((2(11))4), (((11)2)4), (2222).

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Last entries of each row give A006241. Row sums are A281145.

Cf. A003238, A008284, A055277, A063834, A273873, A289501, A294080, A298422, A298426, A299201, A299203, A300442, A300443, A301343, A301364-A301368.

sametrees[n_]:=Prepend[Join@@Table[Tuples[sametrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&&SameQ@@#&]}],n];
Table[Length[Select[sametrees[n],Count[#,_Integer,{-1}]===k&]],{n,12},{k,n}]

A(n)={my(v=vector(n)); for(n=1, n, v[n] = x + sumdiv(n, d, v[n/d]^d)); apply(p -> Vecrev(p/x), v)}
{my(v=A(16)); for(n=1, #v, print(v[n]))} \\ Andrew Howroyd, Aug 20 2018

A348661 a(1) = 1; a(n) = Sum_{d|n, d < n} d * a(d)^(n/d).

Original entry on oeis.org

1, 1, 1, 3, 1, 6, 1, 39, 4, 8, 1, 330, 1, 10, 9, 12495, 1, 1446, 1, 1620, 11, 14, 1, 1792050, 6, 16, 580, 10158, 1, 53002, 1, 2516534175, 15, 20, 13, 469241466, 1, 22, 17, 774558756, 1, 1696170, 1, 712914, 20160, 26, 1, 108457624531554, 8, 328588, 21, 6383964

Offset: 1

Ilya Gutkovskiy, Oct 28 2021

nonn

Cf. A006241, A008578 (positions of 1's), A157313, A165552, A196545, A281145.

a[1] = 1; a[n_] := a[n] = Sum[If[d < n, d a[d]^(n/d), 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 52}]

For n > 1, a(n) is the coefficient of x^n/n in expansion of -log(Product_{k=1..n-1} (1 - a(k)*x^k)).

cp's OEIS Frontend

A281145 Number of same-trees of weight n.

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A297791 Number of series-reduced leaf-balanced rooted trees with n nodes. Number of orderless same-trees with n nodes and all leaves equal to 1.

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

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

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A294019 Number of same-trees whose leaves are the parts of the integer partition with Heinz number n.

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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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