A005804 -id:A005804 - cp's OEIS frontend

A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 2, 2, 4, 6, 12, 8, 28, 20, 32, 38, 112, 76, 116, 58, 352, 236, 1296, 176, 540, 288, 4448, 374, 612, 1144, 1812, 824, 16640, 1316, 59968, 612, 2336, 4528, 3208, 2924, 231168, 18320, 10632, 2168, 856960, 7132, 3334400, 3776, 11684, 74080, 12679424, 4919, 19192

Offset: 1

Gus Wiseman, Sep 04 2018

nonn

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A tree-partition of m is either m itself or a sequence of tree-partitions, one of each part of a multiset partition of m with at least two parts.

			The a(6) = 6 tree-partitions of {1,1,2}:
  (112)
  ((1)(12))
  ((2)(11))
  ((1)(1)(2))
  ((1)((1)(2)))
  ((2)((1)(1)))

Cf. A000311, A001055, A196545, A281118, A281119, A305936, A318762, A318812, A318813, A318846, A318848.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
allmsptrees[m_]:=Prepend[Join@@Table[Tuples[allmsptrees/@p],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Length[allmsptrees[nrmptn[n]]],{n,20}]

a(n) = A281118(A181821(n)).
a(prime(n)) = A289501(n).
a(2^n) = A005804(n).

More terms from Jinyuan Wang, Jun 26 2020

A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 5, 9, 19, 38, 79, 163, 352, 750, 1633, 3558, 7783, 17020, 37338, 81920, 180399, 398600, 885101, 1975638, 4435741, 10013855, 22726109, 51807432, 118545425, 272024659, 625488420, 1440067761, 3317675261, 7644488052, 17610215982, 40547552277, 93298838972, 214516498359, 492844378878

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches, and balanced if all leaves are the same distance from the root. In an identity tree, all branches directly under any given node are different.

			The a(1) = 1 through a(5) = 19 rooted identity trees:
  (1)  (2)   (3)        (4)         (5)
       (11)  (21)       (22)        (32)
             (111)      (31)        (41)
             ((1)(2))   (211)       (221)
             ((1)(11))  (1111)      (311)
                        ((1)(3))    (2111)
                        ((1)(21))   (11111)
                        ((2)(11))   ((1)(4))
                        ((1)(111))  ((2)(3))
                                    ((1)(31))
                                    ((1)(22))
                                    ((2)(21))
                                    ((3)(11))
                                    ((1)(211))
                                    ((11)(21))
                                    ((2)(111))
                                    ((1)(1111))
                                    ((11)(111))
                                    ((1)(2)(11))

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

Cf. A000669, A005804, A048816, A079500, A119262, A120803, A141268, A292504, A300660, A319312.

Cf. A320154, A320155, A320160, A320171, A320173, A320177, A320178, A320179.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gig[m_]:=Prepend[Join@@Table[Union[Sort/@Select[Sort/@Tuples[gig/@mtn],UnsameQ@@#&]],{mtn,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[Select[gig[y],SameQ@@Length/@Position[#,_Integer]&]],{y,Sort /@IntegerPartitions[n]}],{n,8}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(u=vector(n, n, numbpart(n)), v=vector(n)); while(u, v+=u; u=WeighT(u)-u); v} \\ Andrew Howroyd, Oct 25 2018

Terms a(13) and beyond from Andrew Howroyd, Oct 25 2018

A320176 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is a strict integer partition of n.

Original entry on oeis.org

1, 1, 3, 3, 5, 13, 15, 23, 33, 99, 109, 183, 251, 383, 1071, 1261, 2007, 2875, 4291, 5829, 16297, 18563, 30313, 42243, 63707, 85351, 125465, 297843, 356657, 556729, 783637, 1151803, 1564173, 2249885, 2988729, 6803577, 8026109, 12465665, 17124495, 25272841, 33657209

Offset: 1

Gus Wiseman, Oct 07 2018

nonn

Also the number of orderless tree-factorizations of Heinz numbers of strict integer partitions of n.

Also the number of phylogenetic trees on a set of distinct labels summing to n.

			The a(1) = 1 through a(7) = 15 rooted trees:
  (1)  (2)  (3)       (4)       (5)       (6)            (7)
            (21)      (31)      (32)      (42)           (43)
            ((1)(2))  ((1)(3))  (41)      (51)           (52)
                                ((1)(4))  (321)          (61)
                                ((2)(3))  ((1)(5))       (421)
                                          ((2)(4))       ((1)(6))
                                          ((1)(23))      ((2)(5))
                                          ((2)(13))      ((3)(4))
                                          ((3)(12))      ((1)(24))
                                          ((1)(2)(3))    ((2)(14))
                                          ((1)((2)(3)))  ((4)(12))
                                          ((2)((1)(3)))  ((1)(2)(4))
                                          ((3)((1)(2)))  ((1)((2)(4)))
                                                         ((2)((1)(4)))
                                                         ((4)((1)(2)))

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000669, A005804, A008289, A141268, A292504, A300660, A319312, A320171, A320174, A320175, A320177, A320178.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
got[m_]:=Prepend[Join@@Table[Union[Sort/@Tuples[got/@p]],{p,Select[sps[m],Length[#]>1&]}],m];
Table[Length[Join@@Table[got[m],{m,Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,20}]

\\ here S(n) is first n terms of A005804.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n,k)={my(v=vector(n)); for(n=1, n, v[n]=binomial(n+k-1, n) + EulerT(v[1..n])[n]); v}
S(n)={my(M=Mat(vectorv(n, k, b(n,k)))); vector(n, k, sum(i=1, k, binomial(k, i)*(-1)^(k-i)*M[i,k]))}
seq(n)={my(u=S((sqrtint(8*n+1)-1)\2)); [sum(i=1, poldegree(p), polcoef(p,i)*u[i]) | p <- Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))-1)]} \\ Andrew Howroyd, Oct 26 2018

a(n) = Sum_{k>0} A008289(n, k)*A005804(k). - Andrew Howroyd, Oct 26 2018

Terms a(31) and beyond from Andrew Howroyd, Oct 26 2018

A330627 Number of non-isomorphic phylogenetic trees with n nodes.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 4, 5, 9, 14, 24, 39, 69, 116, 205, 357, 632, 1118, 2001, 3576, 6445, 11627, 21080, 38293, 69819, 127539, 233644, 428825, 788832, 1453589, 2683602, 4962167, 9190155, 17044522, 31655676, 58866237, 109600849, 204293047, 381212823, 712073862

Offset: 1

Gus Wiseman, Dec 28 2019

nonn

A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets. Each branching as well as each element of each leaf contributes to the number of nodes.

			Non-isomorphic representatives of the a(2) = 1 through a(9) = 9 trees (commas and outer brackets elided):
  1  12  123  1234    12345    123456     1234567      12345678
              (1)(2)  (1)(23)  (1)(234)   (1)(2345)    (1)(23456)
                               (12)(34)   (12)(345)    (12)(3456)
                               (1)(2)(3)  (1)(2)(34)   (123)(456)
                                          (1)((2)(3))  (1)(2)(345)
                                                       (1)(23)(45)
                                                       (1)((2)(34))
                                                       (1)(2)(3)(4)
                                                       (12)((3)(4))

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

Phylogenetic trees by number of labels are A005804, with unlabeled version A141268.
Balanced phylogenetic trees are A320154.
Cf. A000311, A000669, A001678, A004114, A005121, A007716, A048816, A060356, A330465, A330467, A330469.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[0]); for(n=1, n-1, v=concat(v, EulerT(v)[n] - v[n] + 1)); v} \\ Andrew Howroyd, Jan 02 2021

G.f.: A(x) satisfies A(x) = x*(1/(1-x) - A(x) - 2 + exp(Sum_{k>0} A(x^k)/k)). - Andrew Howroyd, Jan 02 2021

Terms a(11) and beyond from Andrew Howroyd, Jan 02 2021

A005805 Number of phylogenetic trees with n labels.

Original entry on oeis.org

1, 2, 5, 18, 107, 1008, 13113, 214238, 4182487, 94747196, 2440784645, 70431957258, 2249856084803, 78802876705608, 3002702793753489, 123649410977736950, 5471808106109912815, 258948617502187143188, 13049542794706527317597, 697673361673877090147490

Offset: 1

N. J. A. Sloane

Stirling transform of [ 1, 1, 1, 4, 26, 236, ... ] = A000311(n-1).

Series-reduced trees where each leaf is a nonempty subset of the set of n labels. [Christian G. Bower, Dec 15 1999]

Foulds, L. R.; Robinson, R. W. Enumeration of phylogenetic trees without points of degree two. Ars Combin. 17 (1984), A, 169-183. Math. Rev. 85f:05045
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..381 (first 100 terms from Vincenzo Librandi)
N. J. A. Sloane, Transforms
Index entries for sequences related to trees

Cf. A000311, A005804.

stirtr:= proc(p) proc(n) add(p(k) *Stirling2(n,k), k=0..n) end end: b:= proc(n) option remember; if n<=1 then n elif n=2 then 1 else (n+1) *b(n-1) +2*add(binomial(n-1, k) *b(k) *b(n-k), k=2..n-2) fi end:
a:= stirtr(n->`if`(n<2,1, b(n-1))):
seq(a(n), n=1..20);  # Alois P. Heinz, Sep 15 2008

max = 18; a311 = CoefficientList[ InverseSeries[ Series[ 1 + 2x - E^x, {x, 0, max}], x], x]*Range[0, max]!; b[1] = 1; b[k_] := a311[[k]]; a[n_] := Sum[ b[k]*StirlingS2[n, k], {k, 1, n}]; Table[ a[n], {n, 1, max}] (* Jean-François Alcover, Feb 22 2012 *)

From Vaclav Kotesovec, Nov 16 2021: (Start)
E.g.f.: exp(2*x)/4 - (1 + LambertW(-exp(exp(x)/2 - 1)/2))^2.
a(n) ~ 2 * log(2)^(3/2) * n^(n-2) / (exp(n) * (log(2) + log(log(2)))^(n - 3/2)).
(End)

More terms from Christian G. Bower, Dec 15 1999

A319118 Number of multimin tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 8, 1, 2, 2, 24, 1, 6, 1, 8, 2, 2, 1, 42, 2, 2, 6, 8, 1, 8, 1, 112, 2, 2, 2, 38, 1, 2, 2, 42, 1, 8, 1, 8, 8, 2, 1, 244, 2, 6, 2, 8, 1, 24, 2, 42, 2, 2, 1, 58, 1, 2, 8, 568, 2, 8, 1, 8, 2, 8, 1, 268, 1, 2, 6, 8, 2, 8, 1, 244, 24

Offset: 1

Gus Wiseman, Sep 10 2018

nonn

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of multimin tree-factorizations, one of each factor in a multimin factorization of n with at least two factors.

			The a(12) = 8 multimin tree-factorizations:
  12,
  (2*6), (4*3), (6*2), (2*2*3),
  (2*(2*3)), ((2*2)*3), ((2*3)*2).
Or as series-reduced plane trees of multisets:
  112,
  (1,12), (11,2), (12,1), (1,1,2),
  (1,(1,2)), ((1,1),2), ((1,2),1).

Cf. A001055, A005804, A020639, A118376, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A318577, A319119.

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Length[mmftrees[n]],{n,100}]

a(prime^n) = A118376(n).

a(product of n distinct primes) = A005804(n).

A330762 Triangle read by rows: T(n,k) is the number of series-reduced rooted trees whose leaves are multisets of colors with a total of n elements using exactly k colors.

Original entry on oeis.org

1, 2, 2, 4, 12, 8, 11, 67, 114, 58, 30, 376, 1230, 1496, 612, 96, 2174, 12038, 26156, 24570, 8374, 308, 12792, 113028, 389968, 630300, 481284, 140408, 1052, 76972, 1043355, 5363331, 13259870, 17008218, 10930150, 2785906, 3648, 471260, 9574934, 70524256, 250201560, 479284952, 508811114, 282141552, 63830764

Offset: 1

Andrew Howroyd, Dec 29 2019

			Triangle begins:
     1;
     2,     2;
     4,    12,       8;
    11,    67,     114,      58;
    30,   376,    1230,    1496,      612;
    96,  2174,   12038,   26156,    24570,     8374;
   308, 12792,  113028,  389968,   630300,   481284,   140408;
  1052, 76972, 1043355, 5363331, 13259870, 17008218, 10930150, 2785906;
  ...
The T(3,2) = 12 trees are: (122), (112), ((1)(22)), ((1)(12)), ((2)(12)), ((2)(11)), ((1)(2)(2)), ((1)(1)(2)), ((1)((2)(2))), ((1)((1)(2))), ((2)((1)(2))), ((2)((1)(1))).

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

Column 1 is A141268.
Main diagonal is A005804.
Row sums are A330469.
Cf. A330763 (leaves are sets).

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(n+k-1, k-1)]))[n])); v}
M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
{ my(T=M(10)); for(n=1, #T~, print(T[n, 1..n])) }

A330763 Triangle read by rows: T(n,k) is the number of series-reduced rooted trees whose leaves are sets of colors with a total of n elements using exactly k colors.

Original entry on oeis.org

1, 1, 2, 2, 8, 8, 5, 41, 90, 58, 12, 204, 852, 1264, 612, 33, 1046, 7428, 19568, 21510, 8374, 90, 5456, 62682, 262912, 496270, 431040, 140408, 261, 29165, 523167, 3291021, 9520220, 13884960, 9947294, 2785906, 766, 158792, 4358182, 39636784, 165204730, 360421716, 426677440, 259854304, 63830764

Offset: 1

Andrew Howroyd, Dec 29 2019

			Triangle begins:
    1;
    1,     2;
    2,     8,      8;
    5,    41,     90,      58;
   12,   204,    852,    1264,     612;
   33,  1046,   7428,   19568,   21510,     8374;
   90,  5456,  62682,  262912,  496270,   431040,  140408;
  261, 29165, 523167, 3291021, 9520220, 13884960, 9947294, 2785906;
  ...
The T(3,2) = 8 trees are: ((1)(12)), ((2)(12)), ((1)(2)(2)), ((1)(1)(2)), ((1)((2)(2))), ((1)((1)(2))), ((2)((1)(2))), ((2)((1)(1))).

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

Column 1 is A000669.
Main diagonal is A005804.
Row sums are A330764.
Cf. A330762 (leaves are multisets).

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(k,n)]))[n])); v}
M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))} \\ Andrew Howroyd, Dec 29 2019

A320221 Irregular triangle where T(n,k) is the number of unlabeled series-reduced rooted trees with n leaves in which every leaf is at height k, (n>=1, min(1,n-1) <= k <= log_2(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 6, 1, 1, 7, 1, 1, 11, 4, 1, 13, 6, 1, 20, 16, 1, 23, 23, 1, 33, 46, 1, 40, 70, 1, 54, 127, 1, 1, 65, 189, 1, 1, 87, 320, 5, 1, 104, 476, 10, 1, 136, 771, 32, 1, 164, 1145, 63, 1, 209, 1795, 154, 1, 252, 2657, 304, 1, 319, 4091, 656

Offset: 1

Gus Wiseman, Oct 07 2018

			Triangle begins:
  1
  1
  1
  1  1
  1  1
  1  3
  1  3
  1  6  1
  1  7  1
  1 11  4
  1 13  6
  1 20 16
  1 23 23
  1 33 46
  1 40 70
The T(11,3) = 6 rooted trees:
   (((oo)(oo))((oo)(ooooo)))
   (((oo)(oo))((ooo)(oooo)))
   (((oo)(ooo))((oo)(oooo)))
   (((oo)(ooo))((ooo)(ooo)))
  (((oo)(oo))((oo)(oo)(ooo)))
  (((oo)(ooo))((oo)(oo)(oo)))

Andrew Howroyd, Table of n, a(n) for n = 1..1154 (rows 1..200)

Row sums are A120803. Second column is A083751. A regular version is A320179.
Cf. A000669, A005804, A048816, A119262, A120803, A141268, A244925, A319312.
Cf. A316624, A320154, A320155, A320160, A320172, A320173.

qurt[n_]:=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[qurt/@ptn]],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}]];
DeleteCases[Table[Length[Select[qurt[n],SameQ[##,k]&@@Length/@Position[#,{}]&]],{n,10},{k,0,n-1}],0,{2}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
T(n)={my(u=vector(n), v=vector(n), h=1); u[1]=1; while(u, v+=u*h; h*=x; u=EulerT(u)-u); v[1]=x; [Vecrev(p/x) | p<-v]}
{ my(A=T(15)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 09 2020

Terms a(36) and beyond from Andrew Howroyd, Dec 09 2020

Name clarified by Andrew Howroyd, Dec 09 2020

A374384 a(n) = floor(Sum_{k=n^3..(n+1)^3} k^(1/3)).

Original entry on oeis.org

1, 12, 51, 134, 281, 508, 835, 1278, 1857, 2588, 3491, 4582, 5881, 7404, 9171, 11198, 13505, 16108, 19027, 22278, 25881, 29852, 34211, 38974, 44161, 49788, 55875, 62438, 69497, 77068, 85171, 93822, 103041, 112844, 123251, 134278, 145945, 158268, 171267, 184958, 199361

Offset: 0

Amrit Awasthi, Jul 07 2024

Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A000384, A003215, A005804, A014105, A374489.

Table[Floor[Sum[(n^3+k)^(1/3),{k,0,3n^2+3n+1}]],{n,0,40}] (* Stefano Spezia, Jul 07 2024 *)

a(n) = 3*n^3+9*n^2\2+4*n+1; \\ Michel Marcus, Jul 09 2024

a(n) = floor(3*n^3+9*n^2/2+4*n+1).
a(2*n) = 24*n^3 + 18*n^2 + 8*n + 1.
a(2*n-1) = 24*n^3-18*n^2+8*n-2 for n > 0.
a(2*n) = A248575(2*n) + 4*n + 1.
a(2*n-1) = A248575(2*n-1) + 4*n - 2.
From Stefano Spezia, Jul 09 2024: (Start)
G.f.: (1 + 9*x + 17*x^2 + 7*x^3 + 2*x^3)/((1 - x)^4*(1 + x)).
E.g.f.: exp(x)*(1 + 11*x + 14*x^2 + 3*x^3). (End)

cp's OEIS Frontend

A318847 Number of tree-partitions of a multiset whose multiplicities are the prime indices of n.

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A320172 Number of series-reduced balanced rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A320176 Number of series-reduced rooted trees whose leaves are strict integer partitions whose multiset union is a strict integer partition of n.

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A330627 Number of non-isomorphic phylogenetic trees with n nodes.

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A005805 Number of phylogenetic trees with n labels.

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Maple

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A319118 Number of multimin tree-factorizations of n.

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A330762 Triangle read by rows: T(n,k) is the number of series-reduced rooted trees whose leaves are multisets of colors with a total of n elements using exactly k colors.

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