A302939 -id:A302939 - cp's OEIS frontend

A000060 Number of signed trees with n nodes.

1, 2, 3, 10, 27, 98, 350, 1402, 5743, 24742, 108968, 492638, 2266502, 10600510, 50235931, 240882152, 1166732814, 5702046382, 28088787314, 139355139206, 695808554300, 3494391117164, 17641695461662, 89495028762682, 456009893224285, 2332997356507678, 11980753878699716, 61739654456234062, 319188605907760846

Offset: 1

N. J. A. Sloane

If only trees with a degree of each node <= 2 (linear chains) are counted, we obtain A005418. If only trees with a degree of each node <= 3 are counted, we obtain 1, 2, 3, 10, 22, 76, 237, 856, ... If the degree is restricted to <= 4 we obtain 1, 2, 3, 10, 27, 92, 323, 1260, ... - R. J. Mathar, Feb 26 2018

			For n=4 nodes and 3 edges, the unsigned tree has two forms: the star and the linear chain. The star has 4 ways of signing its 3 edges: without plus (3 minus'), with one plus (2 minus'), with two plusses (1 minus) and with three plusses (no minus).  The linear chain has 6 ways of signing the edges: +++, ---, +-- (equivalent to --+), -++ (equivalent to ++-), -+- and +-+. The total number of ways is a(4) = 4+6=10. - _R. J. Mathar_, Feb 26 2018

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..500
F. Harary and G. Prins, The number of homeomorphically irreducible trees and other species, Acta Math., 101 (1959), 141-162.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A000151, A000238.

Row sums of A302939.

unassign('x'): with(combstruct): norootree:=[S, {B = Set(S), S = Prod(Z,B,B)}, unlabeled]: S:=x->add(count(norootree,size=i)*x^i,i=1..30): seq(coeff(S(x)+S(x^2)-S(x)^2,x,i),i=1..29); # with Algolib (Pab Ter)

b[M_] := Module[{A}, A = Table[1, {M}]; For[n = 1, n <= M-1, n++, A[[n+1]] = 2/n*Sum[Sum[d*A[[d]], {d, Divisors[i]}]*A[[n-i+1]], {i, 1, n}]]; A];
seq[n_] := Module[{g}, g = x*(b[n].x^Range[0, n-1]); CoefficientList[g + (g /. x -> x^2) - g^2, x]][[2 ;; n+1]];
seq[29] (* Jean-François Alcover, Sep 04 2019, after Andrew Howroyd *)

\\ here b(N) is A000151 as vector
b(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 2/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); A}
seq(n) = {my(g=x*Ser(b(n))); Vec(g + subst(g, x, x^2) - g^2)} \\ Andrew Howroyd, May 13 2018

G.f.: S(x) + S(x^2) - S(x)^2, where S(x) is the generating function for A000151. - Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005

a(n) = A000238(n) + A000151(n/2), where A000151(.) is zero for non-integer arguments. - R. J. Mathar, Apr 16 2018

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005

A304489 Triangle read by rows: T(n,k) = number of rooted signed trees with n nodes and k positive edges (0 <= k < n).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 9, 9, 4, 9, 26, 37, 26, 9, 20, 75, 134, 134, 75, 20, 48, 214, 469, 596, 469, 214, 48, 115, 612, 1577, 2445, 2445, 1577, 612, 115, 286, 1747, 5204, 9480, 11513, 9480, 5204, 1747, 286, 719, 4995, 16865, 35357, 50363, 50363, 35357, 16865, 4995, 719

Offset: 1

Andrew Howroyd, May 13 2018

Equivalently, the number of rooted trees with 2-colored non-root nodes, n nodes and k nodes of the first color.

			Triangle begins:
    1;
    1,    1;
    2,    3,    2;
    4,    9,    9,    4;
    9,   26,   37,   26,     9;
   20,   75,  134,  134,    75,   20;
   48,  214,  469,  596,   469,  214,   48;
  115,  612, 1577, 2445,  2445, 1577,  612,  115;
  286, 1747, 5204, 9480, 11513, 9480, 5204, 1747, 286;
  ...

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

Row sums are A000151.
Columns k=0..1 are A000081, A000243.
Cf. A294783, A302939, A331114.

R(n, y)={my(v=vector(n)); v[1]=1; for(k=1, n-1, my(p=(1+y)*v[k]); my(q=Vec(prod(j=0, poldegree(p, y), (1/(1-x*y^j) + O(x*x^(n\k)))^polcoeff(p, j)))); v=vector(n, j, v[j] + sum(i=1, (j-1)\k, v[j-i*k] * q[i+1]))); v; }
{ my(A=R(10,y)); for(n=1, #A, print(Vecrev(A[n]))) }

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
R(n, y)={my(v=[1]); for(k=2,n,v=concat([1], EulerMT(v*(1+y)))); v}
{ my(A=R(10,y)); for(n=1, #A, print(Vecrev(A[n]))) }

A331113 Number of signed trees with n positive edges and n negative edges.

Original entry on oeis.org

1, 1, 9, 96, 1439, 24758, 476056, 9841611, 214997756, 4900190330, 115530636036, 2800145556058, 69446136484047, 1756114394736162, 45152546920446518, 1177790435393100588, 31111743336031473800, 831007304851753985293, 22416683345590695488558, 610057933331757664054671

Offset: 0

Andrew Howroyd, Jan 09 2020

nonn

The total number of nodes is 2n + 1. Trees are unrooted.

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A302939, A331114.

a(n) = A302939(2*n + 1, n).

A321305 Triangle T(n,f): the number of signed cubic graphs on 2n vertices with f edges of the first sign.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 2, 1, 1, 2, 3, 8, 16, 21, 21, 16, 8, 3, 2, 5, 14, 57, 152, 313, 474, 551, 474, 313, 152, 57, 14, 5, 19, 91, 491, 1806, 5034, 10604, 17318, 22033, 22033, 17318, 10604, 5034, 1806, 491, 91, 19, 85, 706, 4981, 23791, 84575, 229078, 487020, 825127, 1127783, 1250632, 1127783, 825127, 487020, 229078, 84575, 23791, 4981, 706, 85

Offset: 0

R. J. Mathar, Nov 03 2018

These are connected, undirected, simple cubic graphs where each edge is signed as either "+" or "-". Row n has 1+3n entries, 0<=f<=3n. The column f=0 (1, 0, 1, 2, 5,...) counts the cubic graphs (A002851). The column f=1 (0, 1, 3, 14, 91, 706,...) counts the edge-rooted cubic graphs.

			The triangle starts:
0 vertices: 1
2 vertices: 0,0,0,0
4 vertices: 1,1,2,3,2,1,1
6 vertices: 2,3,8,16,21,21,16,8,3,2
8 vertices: 5,14,57,152,313,474,551,474,313,152,57,14,5
10 vertices: 19,91,491,1806,5034,10604,17318,22033,22033,17318,10604,5034,1806,491,91,19

Cf. A002851 (first column), A321304 (signed vertices), A302939 (signed trees).

T(n,f) = T(n,3*n-f).

cp's OEIS Frontend

A000060 Number of signed trees with n nodes.

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A304489 Triangle read by rows: T(n,k) = number of rooted signed trees with n nodes and k positive edges (0 <= k < n).

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A331113 Number of signed trees with n positive edges and n negative edges.

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A321305 Triangle T(n,f): the number of signed cubic graphs on 2n vertices with f edges of the first sign.

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