A319122 -id:A319122 - cp's OEIS frontend

A319436 Number of palindromic plane trees with n nodes.

1, 1, 2, 3, 6, 10, 20, 35, 68, 122, 234, 426, 808, 1484, 2798, 5167, 9700, 17974, 33656, 62498, 116826, 217236, 405646, 754938, 1408736, 2623188, 4892848, 9114036, 16995110, 31664136, 59034488, 110004243, 205068892, 382156686, 712363344, 1327600346, 2474618434

Offset: 1

Gus Wiseman, Sep 18 2018

nonn

A rooted plane tree is palindromic if the sequence of branches directly under any given node is a palindrome.

			The a(7) = 20 palindromic plane trees:
  ((((((o))))))  (((((oo)))))  ((((ooo))))  (((oooo)))  ((ooooo))  (oooooo)
                 ((((o)(o))))  (((o(o)o)))  ((o(oo)o))  (o(ooo)o)
                 (((o))((o)))  ((o((o))o))  (o((oo))o)  (oo(o)oo)
                               (((o)o(o)))  ((oo)(oo))
                               (o(((o)))o)  ((o)oo(o))
                               ((o)(o)(o))  (o(o)(o)o)

Andrew Howroyd, Table of n, a(n) for n = 1..500
Gus Wiseman, The a(8) = 35 palindromic plane trees.
Gus Wiseman, The a(11) = 234 palindromic plane trees.

Cf. A000108, A000670, A001003, A005043, A008965, A025065, A032128, A118376, A242414, A317085, A317086, A317087, A319122, A319437.

panplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[panplane/@c],#==Reverse[#]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[panplane[n]],{n,10}]

PAL(p)={(1+p)/subst(1-p, x, x^2)}
seq(n)={my(p=O(1));for(i=1, n, p=PAL(x*p)); Vec(p)} \\ Andrew Howroyd, Sep 19 2018

a(n) ~ c * d^n, where d = 1.86383559155190653688720443906758855085492625375... and c = 0.24457511051198663873739022949952908293770055... - Vaclav Kotesovec, Nov 16 2021

A277130 Number of planar branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 6, 2, 3, 1, 14, 1, 3, 3, 24, 1, 14, 1, 14, 3, 3, 1, 78, 2, 3, 6, 14, 1, 25, 1, 112, 3, 3, 3, 110, 1, 3, 3, 78, 1, 25, 1, 14, 14, 3, 1, 464, 2, 14, 3, 14, 1, 78, 3, 78, 3, 3, 1, 206, 1, 3, 14, 568, 3, 25, 1, 14, 3, 25, 1, 850, 1, 3, 14, 14

Offset: 1

Michel Marcus, Oct 01 2016

nonn

A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. - Gus Wiseman, Sep 11 2018

			From _Gus Wiseman_, Sep 11 2018: (Start)
The a(12) = 14 planar branching factorizations:
  12,
  (2*6), (3*4), (4*3), (6*2), (2*2*3), (2*3*2), (3*2*2),
  (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).
(End)

Daniel Mondot, Table of n, a(n) for n = 1..9999
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See page 15.

Cf. A277120.
Cf. A000108, A001055, A074206, A118376, A281113, A319122, A319123.
Cf. A277130, A281118, A281119, A292504, A292505, A295279, A295281, A319136, A319137, A319138.

#include 
#include 
#include 
#define MAX 10000
/* Number of planar branching factorizations of n. */
#define lu unsigned long
lu nbr[MAX]; /* number of branching */
lu a, b, d, e; /* temporary variables */
lu n; lu m, p; // factors of n
lu x; // square root of n
void main(unsigned argc, char *argv[])
{
  memset(nbr, 0, MAX*sizeof(lu));
  for (b=0, n=1; nDaniel Mondot, May 19 2017 */

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];
Table[Length[otfs[n]],{n,20}] (* Gus Wiseman, Sep 11 2018 *)

a(prime^n) = A118376(n). a(product of n distinct primes) = A319122(n). - Gus Wiseman, Sep 11 2018

Terms a(65) and beyond from Daniel Mondot, May 19 2017

A319379 Number of plane trees with n nodes where the sequence of branches directly under any given node is a chain of multisets.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 93, 207, 452, 997, 2176, 4776, 10418, 22781, 49674, 108421

Offset: 1

Gus Wiseman, Sep 17 2018

			The a(6) = 19 chain trees:
  (((((o)))))  ((((oo))))  (((ooo)))  ((oooo))  (ooooo)
               (((o)(o)))  ((o)(oo))  (o(ooo))
               (((o(o))))  ((o(oo)))  (oo(oo))
               ((o((o))))  ((oo(o)))  (ooo(o))
               (o(((o))))  (o((oo)))
                           (o(o)(o))
                           (o(o(o)))
                           (oo((o)))

Gus Wiseman, The a(9) = 207 chain trees.

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A143363, A316470, A319122, A319378, A319380, A319381.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
chnplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[chnplane/@c],And@@submultisetQ@@@Partition[#,2,1]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[chnplane[n]],{n,10}]

A319380 Number of plane trees with n nodes where the sequence of branches directly under any given node is a chain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 17, 30, 53, 94, 169, 303, 543, 968, 1728, 3080, 5491, 9776, 17415, 31008

Offset: 1

Gus Wiseman, Sep 17 2018

			The a(8) = 17 locally identity chain trees:
  (((((((o)))))))  (((((o(o))))))  (((o(o(o)))))  (o(o(o(o))))
                   ((((o((o))))))  ((o((o(o)))))  (o(o)(o(o)))
                   (((o(((o))))))  ((o(o((o)))))
                   ((o((((o))))))  (((o)(o(o))))
                   (o(((((o))))))  (o(((o(o)))))
                                   (o((o((o)))))
                                   (o(o(((o)))))
                                   ((o)(o((o))))
                                   (((o))(o(o)))

Gus Wiseman, The a(12) = 169 identity chain trees.

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A316470, A319122, A319379, A319381.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
idchnplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[idchnplane/@c],And[UnsameQ@@#,And@@submultisetQ@@@Partition[#,2,1]]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[idchnplane[n]],{n,10}]

A319381 Number of plane trees with n nodes where the sequence of branches directly under any given node is a membership-chain.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 9, 11, 20, 28, 40, 58, 82, 110, 159, 217, 305, 420, 570, 767, 1042

Offset: 1

Gus Wiseman, Sep 17 2018

			The a(9) = 11 membership-chain trees:
  ((((((((o))))))))  (((((((o)o))))))  ((((((o)o)o))))  (((((o)o)o)o))
                     ((((((o))(o)))))  (((((o)o)(o))))  ((((o)o)(o)o))
                     (((((o)))((o))))  (((((o))(o)o)))  ((((o))(o)o)o)
                                       ((((o))(o))(o))

Gus Wiseman, The a(15) = 110 membership-chain trees.

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A316470, A319122, A319379, A319380, A319436.

yanplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[yanplane/@c],And@@MemberQ@@@Partition[#,2,1]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[yanplane[n]],{n,10}]

A319123 Number of series-reduced plane trees with n leaves such that each branch directly under any given node has a different number of leaves.

Original entry on oeis.org

1, 1, 3, 7, 21, 75, 277, 1083, 4419, 18493, 77729, 332557, 1444477, 6307225, 27912147, 123878207, 554733045, 2492087531, 11280537097, 51120499279, 233319480419, 1065835004917, 4895443823281, 22505853359485, 103958158302085, 480365303903637, 2229412587062123

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

			The a(4) = 7 plane trees:
  (oooo)
  (o(ooo))
  ((ooo)o)
  (o(o(oo)))
  (o((oo)o))
  ((o(oo))o)
  (((oo)o)o)

Cf. A000108, A001003, A007853, A074206, A118376, A273873, A277130, A281113, A304173, A304175, A319122.

b[n_]:=b[n]=1+Sum[Times@@b/@f,{f,Join@@Permutations/@Select[IntegerPartitions[n],And[Length[#]>1,UnsameQ@@#]&]}];
Array[b,30]

A319137 Number of strict planar branching factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 3, 7, 1, 9, 1, 9, 3, 3, 1, 37, 1, 3, 3, 9, 1, 25, 1, 21, 3, 3, 3, 57, 1, 3, 3, 37, 1, 25, 1, 9, 9, 3, 1, 161, 1, 9, 3, 9, 1, 37, 3, 37, 3, 3, 1, 153, 1, 3, 9, 75, 3, 25, 1, 9, 3, 25, 1, 345, 1, 3, 9, 9, 3, 25, 1, 161

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A strict planar branching factorization of n is either the number n itself or a sequence of at least two strict planar branching factorizations, one of each factor in a strict ordered factorization of n.

			The a(12) = 9 trees:
  12,
  (2*6), (3*4), (4*3),(6*2),
  (2*(2*3)), (2*(3*2)), ((2*3)*2), ((3*2)*2).

Cf. A000108, A001055, A045778, A074206, A118376, A277130, A281113, A281118, A292504, A295279, A295281, A317144, A319122, A319123, A319138.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
sotfs[n_]:=Prepend[Join@@Table[Tuples[sotfs/@f],{f,Select[ordfacs[n],And[Length[#]>1,UnsameQ@@#]&]}],n];
Table[Length[sotfs[n]],{n,100}]

a(prime^n) = A319123(n + 1).

a(product of n distinct primes) = A319122(n).

A319138 Number of complete strict planar branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 4, 1, 2, 2, 0, 1, 4, 1, 4, 2, 2, 1, 8, 0, 2, 0, 4, 1, 18, 1, 0, 2, 2, 2, 28, 1, 2, 2, 8, 1, 18, 1, 4, 4, 2, 1, 16, 0, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 84, 1, 2, 4, 0, 2, 18, 1, 4, 2, 18, 1, 112, 1, 2, 4, 4, 2, 18, 1, 16, 0, 2, 1

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A strict planar branching factorization of n is either the number n itself or a sequence of at least two strict planar branching factorizations, one of each factor in a strict ordered factorization of n. A strict planar branching factorization is complete if the leaves are all prime numbers.

			The a(12) = 4 trees: (2*(2*3)), (2*(3*2)), ((2*3)*2), ((3*2)*2).

Cf. A000108, A045778, A074206, A118376, A277130, A281113, A281118, A295279, A295281, A317144, A319122, A319123, A319136, A319137.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
sotfs[n_]:=Prepend[Join@@Table[Tuples[sotfs/@f],{f,Select[ordfacs[n],And[Length[#]>1,UnsameQ@@#]&]}],n];
Table[Length[Select[sotfs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

a(prime^n) = A000007(n - 1).

a(product of n distinct primes) = A032037(n).

A319378 Number of plane trees with n nodes where the sequence of branches directly under any given node with at least two branches has empty intersection.

Original entry on oeis.org

1, 1, 2, 5, 13, 39, 118, 375, 1225, 4079, 13794, 47287, 163962, 573717, 2023800

Offset: 1

Gus Wiseman, Sep 17 2018

			The a(5) = 13 locally nonintersecting plane trees:
  ((((o))))  (((oo)))  ((ooo))  (oooo)
             (((o)o))  ((oo)o)
             ((o(o)))  (o(oo))
             (((o))o)  ((o)oo)
             (o((o)))  (o(o)o)
                       (oo(o))

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A143363, A316470, A319122, A319379, A319380, A319381, A319436.

monplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[monplane/@c],Or[Length[#]==1,Intersection@@#=={}]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[monplane[n]],{n,10}]

A319136 Number of complete planar branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 9, 1, 2, 2, 11, 1, 9, 1, 9, 2, 2, 1, 44, 1, 2, 3, 9, 1, 18, 1, 45, 2, 2, 2, 66, 1, 2, 2, 44, 1, 18, 1, 9, 9, 2, 1, 225, 1, 9, 2, 9, 1, 44, 2, 44, 2, 2, 1, 132, 1, 2, 9, 197, 2, 18, 1, 9, 2, 18, 1, 450, 1, 2, 9, 9, 2, 18, 1, 225

Offset: 1

Gus Wiseman, Sep 11 2018

nonn

A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. A planar branching factorization is complete if the leaves are all prime numbers.

			The a(12) = 9 trees:
  (2*2*3), (2*3*2), (3*2*2),
  (2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).

Cf. A000108, A001055, A007716, A074206, A118376, A277130, A281118, A281119, A292505, A317144, A319122, A319138.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f],{f,Select[ordfacs[n],Length[#]>1&]}],n];
Table[Length[Select[otfs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

a(prime^n) = A001003(n - 1).

a(product of n distinct primes) = A032037(n).

cp's OEIS Frontend

A319436 Number of palindromic plane trees with n nodes.

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A277130 Number of planar branching factorizations of n.

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A319379 Number of plane trees with n nodes where the sequence of branches directly under any given node is a chain of multisets.

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A319380 Number of plane trees with n nodes where the sequence of branches directly under any given node is a chain of distinct multisets.

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A319381 Number of plane trees with n nodes where the sequence of branches directly under any given node is a membership-chain.

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A319123 Number of series-reduced plane trees with n leaves such that each branch directly under any given node has a different number of leaves.

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A319137 Number of strict planar branching factorizations of n.

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A319138 Number of complete strict planar branching factorizations of n.

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