A245870 -id:A245870 - cp's OEIS frontend

A000151 Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.

1, 2, 7, 26, 107, 458, 2058, 9498, 44947, 216598, 1059952, 5251806, 26297238, 132856766, 676398395, 3466799104, 17873508798, 92630098886, 482292684506, 2521610175006, 13233573019372, 69687684810980, 368114512431638, 1950037285256658, 10357028326495097, 55140508518522726, 294219119815868952, 1573132563600386854, 8427354035116949486, 45226421721391554194

Offset: 1

N. J. A. Sloane

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 286.
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 307 and 564.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 60, R(x).
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
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).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1300 (terms 1..500 from N. J. A. Sloane)
Sølve Eidnes, Order theory for discrete gradient methods, arXiv:2003.08267 [math.NA], 2020.
L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.
Vsevolod Gubarev, Rota-Baxter operators on a sum of fields, arXiv:1811.08219 [math.RA], 2018.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 387
P. Leroux and B. Miloudi, Generalisations de la formule d'Otter, Ann. Sci. Math. Quebec 16 (1992), no 1, 53-80.
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)
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.
R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97.
R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
S. G. Wagner, An identity for the cycle indices of rooted tree automorphism groups, Elec. J. Combinat., 13 (2006), #R00.
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000238, A038055.
Also the self-convolution of A005750. - Paul D. Hanna, Aug 17 2002
Column k=2 of A242249.
Cf. A005751, A245870.

R:=series(x+2*x^2+7*x^3+26*x^4,x,5); M:=500;
for n from 5 to M do
series(add( subs(x=x^k,R)/k, k=1..n-1),x,n);
t4:=coeff(series(x*exp(%)^2,x,n+1),x,n);
R:=series(R+t4*x^n,x,n+1); od:
for n from 1 to M do lprint(n,coeff(R,x,n)); od: # N. J. A. Sloane, Mar 10 2007
with(combstruct):norootree:=[S, {B = Set(S), S = Prod(Z,B,B)}, unlabeled] :seq(count(norootree,size=i),i=1..30); # with Algolib (Pab Ter)

terms = 30; A[] = 0; Do[A[x] = x*Exp[2*Sum[A[x^k]/k, {k, 1, terms}]] + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest
(* Jean-François Alcover, Jun 08 2011, updated Jan 11 2018 *)

seq(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} \\ Andrew Howroyd, May 13 2018

Generating function A(x) = x+2*x^2+7*x^3+26*x^4+... satisfies A(x)=x*exp( 2*sum_{k>=1}(A(x^k)/k) ) [Harary]. - Pab Ter (pabrlos2(AT)yahoo.com), Oct 12 2005
G.f.: x*Product_{n>=1} 1/(1 - x^n)^(2*a(n)) = Sum_{n>=1} a(n)*x^n.
a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.64654261623294971289271351621..., c = 0.2078615974229174213216534920508516879353537904602582293754027908931077971... - Vaclav Kotesovec, Aug 20 2014, updated Dec 26 2020

Extended with alternate description by Christian G. Bower, Apr 15 1998

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

A038055 Number of n-node rooted trees with nodes of 2 colors.

Original entry on oeis.org

2, 4, 14, 52, 214, 916, 4116, 18996, 89894, 433196, 2119904, 10503612, 52594476, 265713532, 1352796790, 6933598208, 35747017596, 185260197772, 964585369012, 5043220350012, 26467146038744, 139375369621960, 736229024863276, 3900074570513316, 20714056652990194

Offset: 1

Christian G. Bower, Jan 04 1999

Alois P. Heinz, Table of n, a(n) for n = 1..1000
L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA] (2018).
R. J. Mathar, Topologically distinct sets of non-intersecting circles in the plane, arXiv:1603.00077 [math.CO] (2016), Table 3.
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A000081, A038056-A038062, A271878 (multisets).

Cf. A245870.

spec := [N, {N=Prod(bead,Set(N)), bead=Union(R,B), R=Atom, B=Atom}]; [seq(combstruct[count](spec, size=n), n=1..40)];
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n<2, 2*n, (add(add(d*
      a(d), d=divisors(j))*a(n-j), j=1..n-1))/(n-1))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, May 11 2014

a[n_] := a[n] = If[n<2, 2*n, (Sum[Sum[d*a[d], {d, Divisors[j]}]*a[n-j], {j, 1, n-1}])/(n-1)]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)
a[1] = 2; a[n_] := a[n] = Sum[k a[k] a[n - m k]/(n-1), {k, n}, {m, (n-1)/k}]; Table[a[n], {n, 30}] (* Vladimir Reshetnikov, Aug 12 2016 *)

seq(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] ) ); 2*A} \\ Andrew Howroyd, May 12 2018

Shifts left and halves under Euler transform.
a(n) = 2*A000151(n).
a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.646542616232949712892713516..., c = 0.41572319484583484264330698410170337587070758092051645875080558178621559423... . - Vaclav Kotesovec, Sep 11 2014, updated Dec 26 2020

A005750 Number of planted matched trees with n nodes.

Original entry on oeis.org

1, 1, 3, 10, 39, 160, 702, 3177, 14830, 70678, 342860, 1686486, 8393681, 42187148, 213828802, 1091711076, 5609297942, 28982708389, 150496728594, 784952565145, 4110491658233, 21602884608167, 113907912618599, 602414753753310, 3194684310627727, 16984594260224529

Offset: 1

N. J. A. Sloane

nonn

When convolved with itself gives A000151.
Number of rooted trees with n nodes and edges not attached to root are 2-colored or oriented.
Also number of 2-trees (with 2n+1 cells) rooted at a symmetric end-edge. - Vladeta Jovovic, Aug 22 2001

			A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 160*x^6 + 702*x^7 + ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.5.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 75, Eq. (3.5.3).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1325 (terms 1..500 from Alois P. Heinz)
Loïc Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 [math.RA], 2018.
T. Fowler, I. Gessel, G. Labelle, P. Leroux, The specification of 2-trees, Adv. Appl. Math. 28 (2) (2002) 145-168, Table 1.
Andrew Gainer-Dewar, Gamma-Species and the Enumeration of k-Trees, Electronic Journal of Combinatorics, Volume 19 (2012), #P45. See page 20, line -3. - From _N. J. A. Sloane_, Dec 15 2012
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 428
R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1991), 93-104.
R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A000151, A058870, A058866, A054581, A245870.

A:= proc(n) option remember; if n=0 then 0 else unapply(convert(series(x*exp(add((A(n-1)(x^k))^2/(k*x^k), k=1..2*n)), x=0,2*n), polynom), x) fi end: a:= n-> coeff(series(A(n)(x), x=0, n+1), x,n): seq(a(n), n=1..23); # Alois P. Heinz, Aug 20 2008

max = 23; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; coes = CoefficientList[ Series[ Log[f[x]/x] - Sum[f[x^k]^2/(k*x^k), {k, 1, max}], {x, 0, max}], x]; eqns = Rest[ Thread[coes == 0]]; s[2] = Solve[eqns[[1]], c[2]][[1]]; Do[eqns = Rest[eqns] /. s[k-1]; s[k] = Solve[ eqns[[1]], c[k]][[1]], {k, 3, max}]; Table[c[k], {k, 1, max}] /. Flatten[ Table[s[k], {k, 2, max}]] (* Jean-François Alcover, Oct 25 2011, after g.f. *)
terms = 26; (* B = g.f. of A000151 *) B[] = 0; Do[B[x] = x*Exp[2*Sum[ B[x^k]/k, {k, 1, terms}]] + O[x]^terms // Normal, terms];
A[x_] = Exp[Sum[B[x^k]/k, {k, 1, terms}]] + O[x]^terms;
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 11 2018 *)

seq(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] ) ); Vec(sqrt(Ser(A)))} \\ Andrew Howroyd, May 13 2018

a(n+1) is Euler transform of A000151.
G.f.: A(x) = x*exp( A(x)^2/x + A(x^2)^2/(2x^2) + A(x^3)^2/(3x^3) + ... + A(x^n)^2/(n*x^n) + ...). [Harary & Palmer (3.5.8)] - Paul D. Hanna
G.f.: sqrt(B(x)/x) where B(x) is the g.f. of A000151. - Andrew Howroyd, May 13 2018
a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.646542616232..., c = 0.06185402386554883780092844840921448929211072031752507960399709674242810089... - Vaclav Kotesovec, Sep 12 2014, updated Dec 26 2020
a(n) = A063687(n)+2*A058870(n). [Harary & Palmer (3.5.3)] - R. J. Mathar, Jan 13 2025

More terms, formula and comment from Christian G. Bower, Dec 15 1999

A242249 Number A(n,k) of rooted trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 7, 4, 0, 0, 1, 4, 15, 26, 9, 0, 0, 1, 5, 26, 82, 107, 20, 0, 0, 1, 6, 40, 188, 495, 458, 48, 0, 0, 1, 7, 57, 360, 1499, 3144, 2058, 115, 0, 0, 1, 8, 77, 614, 3570, 12628, 20875, 9498, 286, 0, 0, 1, 9, 100, 966, 7284, 37476, 111064, 142773, 44947, 719, 0

Offset: 0

Alois P. Heinz, May 09 2014

From Vaclav Kotesovec, Aug 26 2014: (Start)
Column k > 0 is asymptotic to c(k) * d(k)^n / n^(3/2), where constants c(k) and d(k) are dependent only on k. Conjecture: d(k) ~ k * exp(1). Numerically:
d(1)   =   2.9557652856519949747148175... (A051491)
d(2)   =   5.6465426162329497128927135... (A245870)
d(3)   =   8.3560268792959953682760695...
d(4)   =  11.0699628777593263124193026...
d(5)   =  13.7856511100846851989303249...
d(6)   =  16.5022088446930015657112211...
d(7)   =  19.2192613290638657575973462...
d(8)   =  21.9366222112987115910888213...
d(9)   =  24.6541883249893084812976812...
d(10)  =  27.3718979186642404090999595...
d(100) = 272.0126359583480733207362718...
d(101) = 274.7309127032967881125015217...
d(200) = 543.8405620978790523837823296...
d(201) = 546.5588426492458787468860222...
d(101)-d(100) = 2.718276744...
d(201)-d(200) = 2.718280551...
(End)

			Square array A(n,k) begins:
  0,  0,    0,     0,      0,      0,       0,       0, ...
  1,  1,    1,     1,      1,      1,       1,       1, ...
  0,  1,    2,     3,      4,      5,       6,       7, ...
  0,  2,    7,    15,     26,     40,      57,      77, ...
  0,  4,   26,    82,    188,    360,     614,     966, ...
  0,  9,  107,   495,   1499,   3570,    7284,   13342, ...
  0, 20,  458,  3144,  12628,  37476,   91566,  195384, ...
  0, 48, 2058, 20875, 111064, 410490, 1200705, 2984142, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 (2018), doi SIGMA 17 (2021)

Columns k=0-10 give: A063524, A000081, A000151, A006964, A052763, A052788, A246235, A246236, A246237, A246238, A246239.
Rows n=0-3 give: A000004, A000012, A001477, A005449.
Lower diagonal gives A242375.
Cf. A255517, A256064.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
      A(d, k), d=divisors(j))*A(n-j, k)*k, j=1..n-1))/(n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

nn = 10; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; Transpose[ Table[Flatten[ sol = SolveAlways[ 0 == Series[ t[x] - x Product[1/(1 - x^i)^(n a[i]), {i, 1, nn}], {x, 0, nn}], x]; Flatten[{0, Table[a[n], {n, 1, nn}]}] /. sol], {n, 0, nn}]] // Grid (* Geoffrey Critzer, Nov 11 2014 *)
A[n_, k_] := A[n, k] = If[n<2, n, Sum[Sum[d*A[d, k], {d, Divisors[j]}] *A[n-j, k]*k, {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 04 2014, translated from Maple *)

\\ ColGf gives column generating function
ColGf(N,k) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = k/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); x*Ser(A)}
Mat(vector(8, k, concat(0, Col(ColGf(7, k-1))))) \\ Andrew Howroyd, May 12 2018

G.f. for column k: x*Product_{n>=1} 1/(1 - x^n)^(k*A(n,k)). - Geoffrey Critzer, Nov 13 2014

A000238 Number of oriented trees with n nodes.

Original entry on oeis.org

1, 1, 3, 8, 27, 91, 350, 1376, 5743, 24635, 108968, 492180, 2266502, 10598452, 50235931, 240872654, 1166732814, 5702001435, 28088787314, 139354922608, 695808554300, 3494390057212, 17641695461662, 89495023510876, 456009893224285, 2332997330210440

Offset: 1

N. J. A. Sloane

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 286.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 60, r(x).
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 138.
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).

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 350 terms from N. J. A. Sloane)
H. R. Afshar, E. A. Bergshoeff, W. Merbis, Interacting spin-2 fields in three dimensions, arXiv preprint arXiv:1410.6164 [hep-th], 2014-2015, JHEP 2015 (2015) # 040.
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), TOC
R. J. Mathar, Oriented trees A000238
R. Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1991), 93-104.
Index entries for sequences related to trees

Cf. A000060, A000151, A051437 (linear oriented), A334827 (oriented star-like).

Diagonal of A335362.

A:= proc(n) option remember; if n=0 then 0 else unapply(convert(series(x*exp(2* add(A(n-1)(x^k)/k, k=1..n-1)), x=0,n), polynom), x) fi end: a:= n-> coeff(series(A(n+1)(x) *(1-A(n+1)(x)), x=0, n+1), x,n): seq(a(n), n=1..26); # Alois P. Heinz, Aug 20 2008

A[n_][y_] := A[n][y] = If[n == 0, 0, Normal[Series[x*Exp[2*Sum[A[n-1][x^k]/k, {k, 1, n-1}]], {x, 0, n}] /. x -> y]]; a[n_] := SeriesCoefficient[A[n+1][x]*(1-A[n+1][x]), {x, 0, n}]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Feb 12 2014, translated from Maple *)

seq(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] ) ); Vec(Ser(A)-x*Ser(A)^2)} \\ Andrew Howroyd, May 13 2018

G.f. = x+x^2+3*x^3+8*x^4+27*x^5+... = R(x)-R(x)^2, where R(x) = g.f. for A000151.

a(n) ~ c * d^n / n^(5/2), where d = A245870 = 5.64654261623294971289271351621..., c = 0.22571615379282714232305... . - Vaclav Kotesovec, Dec 08 2014

2 errors corrected by Paul Zimmermann, Mar 01 1996

More terms from N. J. A. Sloane, Mar 10 2007

A090381 Expansion of (1+4x+7x^2)/((1-x)^2*(1-x^2)).

Original entry on oeis.org

1, 6, 19, 36, 61, 90, 127, 168, 217, 270, 331, 396, 469, 546, 631, 720, 817, 918, 1027, 1140, 1261, 1386, 1519, 1656, 1801, 1950, 2107, 2268, 2437, 2610, 2791, 2976, 3169, 3366, 3571, 3780, 3997, 4218, 4447, 4680, 4921, 5166, 5419, 5676, 5941, 6210, 6487, 6768, 7057, 7350, 7651, 7956, 8269

Offset: 0

N. J. A. Sloane, Jan 30 2004

Also degree of toric ideal associated with path with n+2 nodes [Eriksson].
Also number of triples (t_1, t_2, t_3) with all t_i in the range 0 <= t_i <= n such that at least one t_i + t_j = n (with i != j). - R. H. Hardin, Aug 04 2014
Conjecture: a(n) is the maximum number of areas that are defined by the 3n angle (n+1)-sectors in a triangle. - Nicolas Nagel, Sep 09 2016

			Some triples for n=10 (from _R. H. Hardin_, Aug 04 2014):
..3....1....2....1....7....9....5....8....5....6....9....4...10....8....6....2
..3....3....8....9....3....3....7....2....9....4....3...10....9....1....8....7
..7....7...10....5....2....1....3....7....1....3....7....0....1....9....4....8

R. H. Hardin and N. J. A. Sloane, Table of n, a(n) for n = 0..1000 [First 210 terms from Hardin]
N. Eriksson, Toric ideals of homogeneous phylogenetic models, arXiv:math/0401175 [math.CO], 2004.
Nicolas Nagel, Example picture for angle (n+1)-sectors
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A090382, A090383, A090384, A090385.
Row 1 of A245869.
Central spine of triangle in A245556. Cf. also A245557.

[3*n*(n+1)+(1+(-1)^n)/2 : n in [0..50]]; // Wesley Ivan Hurt, Sep 13 2016

f:=n-> if n mod 2 = 0 then t:=n/2; 12*t^2+6*t+1 else
t:=(n-1)/2; 12*t^2+18*t+6; fi;
[seq(f(n), n=0..100)];

CoefficientList[Series[(1 + 4 x + 7 x^2)/((1 - x)^2*(1 - x^2)), {x, 0, 52}], x] (* Michael De Vlieger, May 07 2016 *)
Table[3 n (n + 1) + (1 + (-1)^n)/2, {n, 0, 52}] (* or *)
LinearRecurrence[{2, 0, -2, 1}, {1, 6, 19, 36}, 53] (* Michael De Vlieger, Sep 12 2016 *)

x='x+O('x^99); Vec((1+4*x+7*x^2)/((1-x)^2*(1-x^2))) \\ Altug Alkan, May 12 2016

G.f.: (1+4x+7x^2)/((1-x)^2*(1-x^2)).
a(2t) = 12t^2+6t+1, a(2t+1) = 12t^2+18t+6 (t >= 0).
The defining g.f. implies the recurrence a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), an empirical discovery of R. H. Hardin.
a(n) = 3*n*(n+1)+(1+(-1)^n)/2. - Wesley Ivan Hurt, May 06 2016
E.g.f.: 3*x*(2 + x)*exp(x) + cosh(x). - Ilya Gutkovskiy, May 06 2016

Edited by N. J. A. Sloane, Aug 04 2014 (merging the old A090381 and A245870).

A198760 Number of initial spin configurations in two-colored rooted trees with n nodes.

Original entry on oeis.org

2, 8, 32, 136, 596, 2712, 12642, 60234, 291840, 1434184, 7130640, 35807114, 181339236, 925139186, 4750176056, 24528421712, 127294780994, 663591911824, 3473315219722, 18246162722278, 96169600405626, 508413199626078, 2695245063006696, 14324688031734740

Offset: 2

N. J. A. Sloane, Oct 29 2011

nonn

Also the number of two-colored rooted trees that have for a given color of the root at least one nearest neighbor node of the root in the other color. - Martin Paech, Apr 16 2012

G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011

Alois P. Heinz, Table of n, a(n) for n = 2..500
E. Kalinowski and W. Gluza, Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit, arXiv:1106.4938, 2011 (Physical Review B, January 2012).
M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012).

Cf. A000081, A038055, A198761, A225823, A245870.

g:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(t*g(i-1$2, 2)+j-1, j)*g(n-i*j, i-1, t), j=0..n/i)))
    end:
a:= n-> 2*(g(n-1$2, 2) -g(n-1$2, 1)):
seq(a(n), n=2..30);  # Alois P. Heinz, May 12 2014

g[n_, i_, t_] := g[n, i, t] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[t*g[i-1, i-1, 2]+j-1, j]*g[n-i*j, i-1, t], {j, 0, n/i}]]]; a[n_] := 2*(g[n-1, n-1, 2] - g[n-1, n-1, 1]) // FullSimplify; Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Nov 25 2014, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(3/2), where d = A245870 = 5.6465426162329497128927135162..., c = 0.29201514711473716704145008728... . - Vaclav Kotesovec, Sep 12 2014

Terms a(8) and a(9) added by Martin Paech, Apr 16 2012
Term a(10) added by Martin Paech, Jul 30 2013
a(11)-a(25) from Alois P. Heinz, May 12 2014

A005751 Number of matched trees with 2n nodes.

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 180, 701, 2891, 12371, 54564, 246319, 1133602, 5300255, 25119554, 120441076, 583373822, 2851023191, 14044428996, 69677569603, 347904448580, 1747195558582, 8820848574074, 44747514381341, 228004950808983, 1166498678253839, 5990376960443432

Offset: 1

N. J. A. Sloane

nonn

This sequence also describes the number of trees on 2n vertices that are in P-position (a player 2 win) in unrooted UVG (Undirected Vertex Geography). This connection is discussed by Fraenkel, Scheinerman, and Ullman in their paper "Undirected Edge Geography." - Kaitlin Bruegge, Jul 14 2017

			a(3)=2; indeed we have the path P_6 and the tree obtained by identifying one endpoint of each of P_2, P_3, and P_3. - _Emeric Deutsch_, Apr 13 2014

S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 307 and 564.
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..400
Aviezri S. Fraenkel, Edward R. Scheinerman, and Daniel Ullman, Undirected Edge Geography, Theoretical Computer Science, 112, (1993), 371-381.
Indranil Ghosh, Python program for computing this sequence (translated from Maple code)
Rodica Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1991), 93-104.
Rodica Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97. (Annotated scanned copy)
Index entries for sequences related to trees

Cf. A000151 for the rooted version.

Cf. A245870.

with(numtheory): r2:= proc(n) option remember; local m; `if`(n=1, 1, 2/(n-1) *add(r2(m) *add(d*r2(d), d=divisors(n-m)), m=1..n-1)) end: p2:= proc(n) option remember; local m; `if`(n=1, 1, 1/(n-1) *add(p2(m) *add(d*r2(d), d=divisors(n-m)), m=1..n-1)) end: m2:= n-> (r2(n) -add(r2(m) *r2(n-m), m=1..n-1) +`if`(irem(n, 2)=0, r2(n/2), p2((n+1)/2)))/2: seq(m2(n), n=1..30); # Alois P. Heinz, Aug 04 2009

r2[n_] := r2[n] = If[n == 1, 1, 2/(n-1)*Sum[r2[m]*Sum[d*r2[d], {d, Divisors[n-m]}], {m, 1, n-1}]]; p2[n_] := p2[n] = If[n == 1, 1, 1/(n-1)*Sum[p2[m]*Sum[d*r2[d], {d, Divisors[n-m]}], {m, 1, n-1}]]; m2[n_] := (r2[n] - Sum[r2[m]*r2[n-m], {m, 1, n-1}] + If[Mod[n, 2] == 0, r2[n/2], p2[(n+1)/2]])/2; Table[m2[n], {n, 1, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(5/2), where d = A245870 = 5.646542616232949712892713..., c = 0.1128580768964135711615258... . - Vaclav Kotesovec, Aug 25 2014

More terms from Alois P. Heinz, Aug 04 2009

cp's OEIS Frontend

A000151 Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.

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A038055 Number of n-node rooted trees with nodes of 2 colors.

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A005750 Number of planted matched trees with n nodes.

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A242249 Number A(n,k) of rooted trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A000238 Number of oriented trees with n nodes.

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A090381 Expansion of (1+4x+7x^2)/((1-x)^2*(1-x^2)).

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