A007853 -id:A007853 - cp's OEIS frontend

A118376 Number of all trees of weight n, where nodes have positive integer weights and the sum of the weights of the children of a node is equal to the weight of the node.

1, 2, 6, 24, 112, 568, 3032, 16768, 95200, 551616, 3248704, 19389824, 117021824, 712934784, 4378663296, 27081760768, 168530142720, 1054464293888, 6629484729344, 41860283723776, 265346078982144, 1687918305128448, 10771600724946944, 68941213290561536

Offset: 1

Jeremy Johnson (jjohnson(AT)cs.drexel.edu), May 15 2006

nonn

The number of trees with leaf nodes equal to 1 is counted by the sequence A001003 of super-Catalan numbers. The number of binary trees is counted by the sequence A007317 and the number of binary trees with leaf nodes equal to 1 is counted by the sequence A000108 of Catalan numbers.
Also the number of series-reduced enriched plane trees of weight n. A series-reduced enriched plane tree of weight n is either the number n itself or a finite sequence of at least two series-reduced enriched plane trees, one of each part of an integer composition of n. For example, the a(3) = 6 trees are: 3, (21), (12), (111), ((11)1), (1(11)). - Gus Wiseman, Sep 11 2018
Conjectured to be the number of permutations of length n avoiding the partially ordered pattern (POP) {1>2, 1>3, 3>4, 3>5} of length 5. That is, conjectured to be the number of length n permutations having no subsequences of length 5 in which the first element is the largest, and the third element is larger than the fourth and fifth elements. - Sergey Kitaev, Dec 13 2020
This conjecture has been proven. It can be restated as the number of size n permutations avoiding 51423, 51432, 52413, 52431, 53412, 53421, 54312, 54321. There are twelve sets of permutations avoiding eight size five permutations that are known to match this sequence. A further four are conjectured to match this sequence. - Christian Bean, Jul 24 2024

			T(3) = 6 because there are six trees
  3    3      3     3     3       3
      2 1    2 1   1 2   1 2    1 1 1
            1 1           1 1
From _Gus Wiseman_, Sep 11 2018: (Start)
The a(4) = 24 series-reduced enriched plane trees:
  4,
  (31), (13), (22), (211), (121), (112), (1111),
  ((21)1), ((12)1), (1(21)), (1(12)), (2(11)), ((11)2),
  ((111)1), (1(111)), ((11)(11)), ((11)11), (1(11)1), (11(11)),
  (((11)1)1), ((1(11))1), (1((11)1)), (1(1(11))).
(End)

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, PermPAL database.
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.
Hadrien Cambazard and Nicolas Catusse, Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the Plane, arXiv preprint arXiv:1512.06649 [cs.DS], 2015.
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Pawel Hitczenko, Jeremy R. Johnson, and Hung-Jen Huang, Distribution of a class of divide and conquer recurrences arising from the computation of the Walsh-Hadamard transform, Theoretical Computer Science, Vol. 352, 2006, pp. 8-30.
J. R. Johnson and M. Püschel, In search for the optimal Walsh-Hadamard transform, Proc. ICASSP, Vol. 4, 2000, pp. 3347-3350.
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A000108, A000311, A001003, A005804, A007317, A007853, A032171, A032200, A143363, A289501, A317852.

T := proc(n) option remember; local C, s, p, tp, k, i; if n = 1 then return 1; else s := 1; for k from 2 to n do C := combinat[composition](n,k); for p in C do tp := map(T,p); s := s + mul(tp[i],i=1..nops(tp)); end do; end do; end if; return s; end;

Rest[CoefficientList[Series[(Sqrt[1-8*x+8*x^2]-1)/(4*x-4), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 03 2014 *)
a[n_] := 1+Sum[Binomial[n-1, k-1]*Hypergeometric2F1[2-k, k+1, 2, -1], {k, 2, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Apr 03 2015, after Vladimir Kruchinin *)
urp[n_]:=Prepend[Join@@Table[Tuples[urp/@ptn],{ptn,Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[urp[n]],{n,7}] (* Gus Wiseman, Sep 11 2018 *)

a(n):=sum((-1)^j*2^(n-j-1)*binomial(n,j)*binomial(2*n-2*j-2,n-2*j-1),j,0,(n-1)/2)/n; /* Vladimir Kruchinin, Sep 29 2020 */

x='x+O('x^25); Vec((sqrt(1-8*x+8*x^2) - 1)/(4*x-4)) \\ G. C. Greubel, Feb 08 2017

Recurrence: T(1) = 1; For n > 1, T(n) = 1 + Sum_{n=n1+...+nt} T(n1)*...*T(nt).
G.f.: (-1+(1-8*z+8*z^2)^(1/2))/(-4+4*z).
From Vladimir Kruchinin, Sep 03 2010: (Start)
O.g.f.: A(x) = A001003(x/(1-x)).
a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A001003(k), n>0. (End)
D-finite with recurrence: n*a(n) + 3*(-3*n+4)*a(n-1) + 4*(4*n-9)*a(n-2) + 8*(-n+3)*a(n-3) = 0. - R. J. Mathar, Sep 27 2013
a(n) ~ sqrt(sqrt(2)-1) * 2^(n-1/2) * (2+sqrt(2))^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 03 2014
From Peter Bala, Jun 17 2015: (Start)
With offset 0, binomial transform of A001003.
O.g.f. A(x) = series reversion of x*(2*x - 1)/(2*x^2 - 1); 2*(1 - x)*A^2(x) - A(x) + x = 0.
A(x) satisfies the differential equation (x - 9*x^2 + 16*x^3 - 8*x^4)*A'(x) + x*(3 - 4*x)*A(x) + x*(2*x - 1) = 0. Extracting coefficients gives Mathar's recurrence above. (End)
a(n) = Sum_{j=0..(n-1)/2} (-1)^j*2^(n-j-1)*C(n,j)*C(2*n-2*j-2,n-2*j-1)/n. - Vladimir Kruchinin, Sep 29 2020

A083884 a(n) = (3^(2*n) + 1) / 2.

Original entry on oeis.org

1, 5, 41, 365, 3281, 29525, 265721, 2391485, 21523361, 193710245, 1743392201, 15690529805, 141214768241, 1270932914165, 11438396227481, 102945566047325, 926510094425921, 8338590849833285, 75047317648499561, 675425858836496045, 6078832729528464401

Offset: 0

Paul Barry, May 09 2003

Number of compositions of even natural numbers into n parts <= 8. - Adi Dani, May 28 2011

a(n) for n >= 1 gives the number of line segments in the n-th iteration of the Peano curve given by plotting (A163528, A163529) or by (Siromoney 1982) when parallel line segments that are connected end-to-end are counted as a single line segment. - Jason V. Morgan, Oct 08 2021

			From _Adi Dani_, May 28 2011: (Start)
a(2)=41: there are 41 compositions of even natural numbers into 2 parts <=8:
(0,0);
(0,2),(2,0),(1,1);
(0,4),(4,0),(1,3),(3,1),(2,2);
(0,6),(6,0),(1,5),(5,1),(2,4),(4,2),(3,3);
(0,8),(8,0),(1,7),(7,1),(2,6),(6,2),(3,5),(5,3),(4,4);
(2,8),(8,2),(3,7),(7,3),(4,6),(6,4),(5,5);
(4,8),(8,4),(5,7),(7,5),(6,6);
(6,8),(8,6),(7,7);
(8,8).  (End)

Siromoney, R., & Subramanian, K.G. (1982). Space-filling curves and infinite graphs. Graph-Grammars and Their Application to Computer Science.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Roberto Amato, A note on Pythagorean Triples, arXiv:1912.05925 [math.HO], 2019. See Example 2.1 p. 4.
Index entries for linear recurrences with constant coefficients, signature (10,-9).

Cf. A000364, A002438, A007853, A083885, A086645.

[(3^(2*n) + 1) / 2: n in [0..20]]; // Vincenzo Librandi, Jun 16 2011

f[n_] := (3^(2n)+1)/2; Table[f@i, {i,0,20}] (* Michael De Vlieger, Jan 28 2015 *)

a(n)=(3^(2*n)+1)/2 \\ Charles R Greathouse IV, Sep 24 2015

a(0) = 1, a(n) = 9*a(n-1) - 4.
a(n) = Sum_{k=0..n} binomial(2*n, 2*k)*4^k.
a(n) = A002438(n) / A000364(n); A000364(n) : Euler numbers.
G.f.: (1-5*x)/((1-x)*(1-9*x)).
a(n) = (3^n + 1^n + (-1)^n + (-3)^n)/4.
E.g.f.: exp(3*x) + exp(x) + exp(-x) + exp(-3*x).
Each term expresses a Pythagorean relationship, along with (a(n)-1) and a power of 3, n>0, such that sqrt((a(n))^2 - (a(n)-1)^2) = 3^n. E.g., 365^2 - 364^2 - 3^3 = 27 (the Pythagorean triangle (365, 364, 27)). - Gary W. Adamson, Jun 25 2006
a(n) = 10*a(n-1) - 9*a(n-2). - Wesley Ivan Hurt, Apr 21 2021

Additional comments from Philippe Deléham, Jul 10 2005

A032200 Number of rooted compound windmills (mobiles) of n nodes.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 51, 128, 345, 940, 2632, 7450, 21434, 62174, 182146, 537369, 1596133, 4767379, 14312919, 43162856, 130695821, 397184252, 1211057426, 3703794849, 11358759346, 34923477315, 107627138308, 332404636811

Offset: 1

Christian G. Bower

Also the number of locally necklace plane trees with n nodes, where a plane tree is locally necklace if the sequence of branches directly under any given node is lexicographically minimal among its cyclic permutations. - Gus Wiseman, Sep 05 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(5) = 9 locally necklace plane trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  (o((o)))
  ((o)(o))
  ((ooo))
  (o(oo))
  (oo(o))
  (oooo)
(End)

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.84).

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. G. Bower, Transforms (2)
Index entries for sequences related to mobiles

Cf. A029768, A038037, A055340.

Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
neckplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[neckplane/@c],neckQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[neckplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)

CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CIK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left under "CIK" (necklace, indistinct, unlabeled) transform.

A102403 Number of Dyck paths of semilength n having no ascents of length 2.

Original entry on oeis.org

1, 1, 1, 2, 6, 17, 46, 128, 372, 1109, 3349, 10221, 31527, 98178, 308179, 973911, 3096044, 9894393, 31770247, 102444145, 331594081, 1077022622, 3509197080, 11466710630, 37567784437, 123380796192, 406120349756, 1339571374103

Offset: 0

Emeric Deutsch, Jan 06 2005

nonn

Number of Łukasiewicz paths of length n having no (1,1) steps. A Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: a(4)=6 because we have HHHH, HU(2)DD, U(2)DDH, U(2)DHD, U(2)HDD and U(3)DDD where H=(1,0), U(2)=(1,2), U(3)=(1,3) and D=(1,-1).

Also the number of series-reduced Mathematica expressions with one atom and n + 1 positions. Also the number of rooted plane trees with n + 1 nodes in which there are no binary branchings (nodes of out-degree 2). - Gus Wiseman, Aug 14 2018

			a(3)=2 because we have UDUDUD and UUUDDD, where U=(1,1) and D=(1,-1); the other three Dyck paths of semilength 3, namely UD(UU)DD, (UU)DDUD and (UU)DUDD, have ascents of length 2 (shown between parentheses).
From _Gus Wiseman_, Aug 14 2018: (Start)
The a(5) = 17 series-reduced Mathematica expressions:
  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[o,o,o][]
  o[][o,o][]
  o[o,o][][]
  o[][][][][]
The a(5) = 17 rooted plane trees with no binary branchings:
  (((((o)))))
  (((ooo)))
  (((o)oo))
  ((o(o)o))
  ((oo(o)))
  ((oooo))
  (((o))oo)
  (o((o))o)
  (oo((o)))
  ((o)(o)o)
  ((o)o(o))
  (o(o)(o))
  ((o)ooo)
  (o(o)oo)
  (oo(o)o)
  (ooo(o))
  (ooooo)
(End)

Robert Israel, Table of n, a(n) for n = 0..1700
Eric S. Egge and Kailee Rubin, Snow Leopard Permutations and Their Even and Odd Threads, arXiv:1508.05310 [math.CO], 2015.
Index entries for sequences related to Łukasiewicz

Cf. A000108, A001003, A001006, A007853, A102402, A126120, A318049.

Order:=35: S:=solve(series(V*(1-V)/(1-V^2+V^3),V)=z,V): seq(coeff(S,z^n),n=1..33); # V=zG
P:= gfun:-rectoproc({(69*n^2+207*n+138)*a(n)+(97*n^2+609*n+830)*a(n+1)+(-38*n^2-342*n-694)*a(n+2)+(37*n^2+333*n+734)*a(n+3)+(2*n^2+18*n+34)*a(n+4)+(-7*n^2-87*n-266)*a(n+5)+(n^2+15*n+56)*a(n+6), a(0)=1,a(1)=1,a(2)=1,a(3)=2,a(4)=6,a(5)=17},a(n),remember):
seq(P(n),n=0..50); # Robert Israel, Aug 24 2015

a[n_] := 1/(n+1) Sum[Binomial[n+1, j] Binomial[3j-n-3, j-1] (-1)^(n+1-j), {j, n+1, (n+3)/3, -1}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 21 2018, after Vladimir Kruchinin *)

a102403(n):=1/n*sum(binomial(n,j)*binomial(3*j-n-2,j-1)*(-1)^(n-j),j,ceiling((n+2)/3),n); /* Vladimir Kruchinin, Mar 07 2011 */

Vec( serreverse( O(x^33) + x/(1+x/(1-x)-x^2) ) /x )  \\ Joerg Arndt, Apr 28 2016

G.f.: G=G(z) satisfies z^3*G^3 + z(1-z)G^2 - G + 1 = 0.
a(n) = (1/n)*Sum_{j=ceiling((n+2)/3)..n} binomial(n,j)*binomial(3*j-n-2,j-1)*(-1)^(n-j), n > 0. - Vladimir Kruchinin, Mar 07 2011
From Gary W. Adamson, Jan 30 2012: (Start)
a(n) is the upper left term in M^n, M = an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  0, 1, 1, 0, 0, 0, ...
  1, 0, 1, 1, 0, 0, ...
  1, 1, 0, 1, 1, 0, ...
  1, 1, 1, 0, 1, 1, ... (End)
(69*n^2+207*n+138)*a(n) + (97*n^2+609*n+830)*a(n+1) + (-38*n^2-342*n-694)*a(n+2) + (37*n^2+333*n+734)*a(n+3) + (2*n^2+18*n+34)*a(n+4) + (-7*n^2-87*n-266)*a(n+5) + (n^2+15*n+56)*a(n+6) = 0. - Robert Israel, Aug 24 2015
Recurrence (of order 4): (n+1)*(n+2)*(28*n^2 - 32*n - 39)*a(n) = 4*(n+1)*(14*n^3 - 9*n^2 - 62*n + 39)*a(n-1) + (140*n^4 - 160*n^3 - 401*n^2 + 469*n - 78)*a(n-2) - 12*(n-2)*(14*n^3 - 9*n^2 - 28*n - 8)*a(n-3) + 23*(n-3)*(n-2)*(28*n^2 + 24*n - 43)*a(n-4). - Vaclav Kotesovec, Mar 06 2016
a(n) ~ s * sqrt((1 - 2*r + 3*r^2*s)/(1 - r + 3*r^2*s)) /(2*sqrt(Pi)*n^(3/2)*r^n), where r = 0.2869905464691794898... and s = 1.850270202250705342... are roots of the system of equations 3*r^3*s^2 = 1 + 2*(-1 + r)*r*s, 1 + r^3*s^3 = s + (-1 + r)*r*s^2. - Vaclav Kotesovec, Mar 06 2016

A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 23, 59, 148, 385, 1006, 2678, 7170, 19421, 52933, 145364, 401421, 1114713, 3109710, 8713076, 24506121, 69168705, 195849114, 556165311, 1583601840, 4520226558, 12931917204, 37075154703

Offset: 1

Christian G. Bower

Also the number of locally Lyndon plane trees with n nodes, where a plane tree is locally Lyndon if the sequence of branches directly under any given node is a Lyndon word. - Gus Wiseman, Sep 05 2018

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(6) = 10 locally Lyndon plane trees:
  (((((o)))))
  (((o(o))))
  ((o((o))))
  (o(((o))))
  ((o)((o)))
  ((oo(o)))
  (o(o(o)))
  (oo((o)))
  (o(o)(o))
  (ooo(o))
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Wikipedia, Lyndon word
Index entries for sequences related to mobiles

Cf. A032200, A055363.

Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

T[n_, k_] := Module[{A}, A[, ] = 0; If[k < 1 || k > n, 0, For[j = 1, j <= n, j++, A[x_, y_] = x*y - x*Sum[MoebiusMu[i]/i * Log[1 -  A [x^i, y^i]] + O[x]^j // Normal , {i, 1, j}]]; Coefficient[Coefficient[A[x, y], x, n], y, k]]];
a[n_] := a[n] = Sum[T[n, k], {k, 1, n}];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 28}] (* Jean-François Alcover, Jun 30 2017, using Michael Somos' code for A055363 *)
LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
lynplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lynplane/@c],LyndonQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lynplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)

CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CHK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Shifts left under "CHK" (necklace, identity, unlabeled) transform.
From Petros Hadjicostas, Dec 03 2017: (Start)
a(n+1) = (1/n)*Sum_{d|n} mu(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = c(1) = 1.
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+1)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)).
The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 147*x^6 + 414*x^7 + 1203*x^8 + ...
(End)

A303022 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and no unitary parts (subexpressions of the form x[y]).

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 27, 63, 152, 376, 939, 2371, 6047, 15577, 40429, 105637, 277625, 733518, 1947126, 5190503, 13888811, 37291968, 100444019, 271316998, 734802247, 1994873116, 5427893149, 14799525982, 40429761365, 110645688034, 303316712450, 832799212777

Offset: 1

Gus Wiseman, Aug 15 2018

nonn

Also the number of orderless Mathematica expressions with one atom, n positions, and no unitary parts.

			The a(6) = 12 Mathematica expressions:
  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[][][][][]

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

Cf. A000108, A001003, A001006, A007853, A102403, A126120, A318049.

Cf. A303023, A303024, A303025, A303026, A303027.

allOLBF[n_]:=allOLBF[n]=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allOLBF[h],Select[Union[Sort/@Tuples[allOLBF/@p]],Length[#]!=1&]}],{p,IntegerPartitions[g]}]]];
Table[Length[allOLBF[n]],{n,10}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)-v); v=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018

A303027 Number of free pure symmetric multifunctions with one atom, n positions, and no empty or unitary parts (subexpressions of the form x[] or x[y]).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 5, 7, 15, 28, 47, 90, 175, 319, 607, 1181, 2251, 4325, 8449, 16425, 31992, 62823, 123521, 243047, 480316, 951290, 1886293, 3749341, 7467815, 14893500, 29752398, 59532947, 119274491, 239275400, 480638121, 966571853, 1945901716, 3921699524

Offset: 1

Gus Wiseman, Aug 15 2018

nonn

Also the number of orderless Mathematica expressions with one atom, n positions, and no empty or unitary parts.

			The a(10) = 15 Mathematica expressions:
  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,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,o,o,o][o,o,o]
  o[o,o,o,o,o][o,o]
  o[o,o,o,o,o,o,o,o]

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

Cf. A000108, A001003, A001006, A007853, A102403, A126120, A318049.

Cf. A303022, A303023, A303024, A303025, A303026.

allOLZR[n_]:=allOLZR[n]=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allOLZR[h],Select[Union[Sort/@Tuples[allOLZR/@p]],Length[#]>1&]}],{p,IntegerPartitions[g]}]]];
Table[Length[allOLZR[n]],{n,25}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)-v); v=concat(v, sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Terms a(29) and beyond from Andrew Howroyd, Aug 19 2018

A304173 Number of rooted plane trees where every branch that has a predecessor (a branch directly to its left and emanating from the same root) has at least as many leaves as its predecessor.

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 90, 242, 660, 1822, 5085, 14333, 40759, 116817, 337140, 979098, 2859439, 8393113, 24747052, 73262246, 217681621, 648939319, 1940461444, 5818595438, 17492367097, 52712114792, 159193762250, 481754196170, 1460650624068, 4436422703787, 13496947320929

Offset: 1

Gus Wiseman, Aug 16 2018

nonn

			The a(5) = 13 plane trees:
  ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
  (((o))o), (o((o))), (o(oo)), ((o)(o)),
  ((o)oo), (o(o)o), (oo(o)),
  (oooo).
Missing from this list is ((oo)o).

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

Cf. A000081, A000108, A001003, A001006, A003238, A007853, A126120, A291442, A291443, A304175.

pplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[pplane/@c],OrderedQ[Count[#,{},{0,Infinity}]&/@#]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[pplane[n]],{n,10}]

seq(n)={my(p=x*y+O(x^2)); for(n=2, n, p=x*(y-1 + 1/prod(k=1, n-1, 1 - y^k*polcoef(p,k,y)))); Vec(subst(p,y,1))} \\ Andrew Howroyd, Jan 22 2021

G.f.: A(x,1) where A(x,y) satisfies A(x,y) = x*(y-1 + 1/(Product_{k>=1} 1 - y^k * [y^k] A(x,y))). - Andrew Howroyd, Jan 22 2021

Terms a(15) and beyond from Andrew Howroyd, Jan 22 2021

A304175 Number of leaf-balanced rooted plane trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 59, 128, 277, 597, 1280, 2730, 5794, 12248, 25836, 54508, 115222, 244144, 518104, 1099499, 2330326, 4930089, 10415135, 21992400, 46470911, 98353146, 208580686, 443186181, 942988423, 2007981801, 4276830431, 9109431322, 19404918449, 41357252072, 88236092543

Offset: 1

Gus Wiseman, Aug 16 2018

nonn

A rooted plane tree is leaf-balanced if every branch of the root has the same number of leaves, and every branch of the root is itself leaf-balanced.

			The a(5) = 12 leaf-balanced plane trees:
  ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
  (((o))o), (o((o))), ((o)(o)),
  ((o)oo), (o(o)o), (oo(o)),
  (oooo).
Missing from this list are ((oo)o) and (o(oo)).

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

Cf. A000081, A000108, A001003, A001006, A003238, A007853, A126120, A291442, A291443, A304173.

lbplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lbplane/@c],SameQ@@(Count[#,{},{0,Infinity}]&/@#)&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lbplane[n]],{n,10}]

seq(n)={my(v=vector(n)); v[1]=x/(1-x) + O(x*x^n); for(k=2, n, v[k]=x*sumdiv(k, d, if(dAndrew Howroyd, Dec 13 2020

Terms a(17) and beyond from Andrew Howroyd, Dec 13 2020

A318049 Number of first/rest balanced rooted plane trees with n nodes.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 3, 2, 6, 8, 11, 26, 28, 67, 96, 162, 316, 448, 922, 1435, 2572, 4660, 7563, 14397, 23896, 43337, 77097, 133071, 244787, 423093, 767732, 1367412, 2426612, 4408497, 7802348, 14152342, 25365035, 45602031, 82631362, 148246136, 269103870, 485379304

Offset: 1

Gus Wiseman, Aug 13 2018

nonn

A rooted plane tree is first/rest balanced if either (1) it is a single node, or (2a) the number of leaves in the first branch is equal to the number of branches minus one, and (2b) every branch is also first/rest balanced.

Also the number of composable free pure multifunctions (CPMs) with one atom and n positions. A CPM is either (case 1) the leaf symbol "o", or (case 2) an expression of the form h[g_1, ..., g_k] where h and each of the g_i for i = 1, ..., k > 0 are CPMs, and the number of leaves in h is equal to k. The number of positions in a CPM is the number of brackets [...] plus the number of o's.

			The a(12) = 11 first/rest balanced rooted plane trees:
  (o(o(o((oo)oo))))
  (o(o((oo)(oo)o)))
  (o(o((oo)o(oo))))
  (o((oo)(o(oo))o))
  (o((oo)o(o(oo))))
  (o((oo)(oo)(oo)))
  ((oo)(o(o(oo)))o)
  ((oo)o(o(o(oo))))
  ((o(o(oo)))oooo)
  ((oo)(o(oo))(oo))
  ((oo)(oo)(o(oo)))
The a(12) = 11 composable free pure multifunctions:
  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[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]

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000081, A000108, A001003, A001006, A007853, A126120, A317713, A318046, A318048.

balplane[n_]:=balplane[n]=If[n===1,{{}},Join@@Function[c,Select[Tuples[balplane/@c],Length[Cases[#[[1]],{},{0,Infinity}]]==Length[#]-1&]]/@Join@@Permutations/@IntegerPartitions[n-1]];
Table[Length[balplane[n]],{n,10}]

seq(n)={my(p=x*y+O(x^2)); for(n=1, n\2, p = x*y + x*sum(k=1, n, y^k * polcoef(p,k,y) * (O(x^(2*n-k+1)) + p)^k )); Vec(subst(p + O(x*x^n), y, 1)) } \\ Andrew Howroyd, Jan 22 2021

G.f.: A(x,1) where A(x,y) satisfies A(x,y) = x*(y + Sum_{k>=1} y^k * ([y^k] A(x,y)) * A(x,y)^k). - Andrew Howroyd, Jan 22 2021

Terms a(21) and beyond from Andrew Howroyd, Jan 22 2021

cp's OEIS Frontend

A118376 Number of all trees of weight n, where nodes have positive integer weights and the sum of the weights of the children of a node is equal to the weight of the node.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A083884 a(n) = (3^(2*n) + 1) / 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A032200 Number of rooted compound windmills (mobiles) of n nodes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A102403 Number of Dyck paths of semilength n having no ascents of length 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A303022 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and no unitary parts (subexpressions of the form x[y]).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI