A005750 -id:A005750 - 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

A005753 Number of rooted identity matched trees with n nodes.

Original entry on oeis.org

1, 2, 5, 18, 66, 266, 1111, 4792, 21124, 94888, 432415, 1994828, 9296712, 43706722, 207030398, 987130456, 4733961435, 22819241034, 110500644857, 537295738556, 2622248720234, 12840953621208, 63074566121245, 310693364823376, 1534374047239554, 7595642577152762

Offset: 1

N. J. A. Sloane

Also number of rooted identity trees with n nodes and 2-colored non-root nodes. - Christian G. Bower, Apr 15 1998

			G.f.: A(x) = x + 2*x^2 + 5*x^3 + 18*x^4 + 66*x^5 + 266*x^6 + ...
where A(x) = x*(1+x)^2*(1+x^2)^4*(1+x^3)^10*(1+x^4)^36*(1+x^5)^132*... (the exponents are A038077(n), n>=1).

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
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 429
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)
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A038077, A246312.

Column k=2 of A255517.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2*a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n=1, 1, b((n-1)$2)):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 01 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[2*a[i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n == 1, 1, b[n-1, n-1]]; Table[a[n] // FullSimplify, {n, 1, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

{a(n)=polcoeff(x*prod(k=1, n-1, (1+x^k+x*O(x^n))^(2*a(k))), n)} /* Paul D. Hanna */

G.f.: x*Product_{n>=1} (1 + x^n)^(2*a(n)) = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Dec 31 2011
a(n) ~ c * d^n / n^(3/2), where d = A246312 = 5.249032491228170579164952216..., c = 0.192066288645200371237879149260484794708740197522264442948290580404909605849... - Vaclav Kotesovec, Aug 25 2014, updated Dec 26 2020
G.f. A(x) satisfies: A(x) = x*exp(2*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k). - Ilya Gutkovskiy, Apr 13 2019

A005754 Number of planted identity matched trees with n nodes.

Original entry on oeis.org

1, 1, 2, 7, 24, 95, 388, 1650, 7183, 31965, 144502, 662241, 3068942, 14358678, 67729973, 321759461, 1538076291, 7392775328, 35707198905, 173221206284, 843634142771, 4123376617009, 20218897206392, 99436453714990, 490355165178472, 2424146632435852

Offset: 1

N. J. A. Sloane

Number of rooted identity trees with n nodes and edges not attached to root are 2-colored or oriented. - Christian G. Bower, Dec 15 1999

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
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 430
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. A005753, A102755, A246312.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2*b((i-1)$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(b((i-1)$2), j)*g(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> g((n-1)$2):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 01 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[2*b[i-1, i-1], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i-1, i-1], j]*g[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := g[n-1, n-1]; Table[a[n], {n, 1, 30}] // FullSimplify (* Jean-François Alcover, Dec 02 2013, translated from Alois P. Heinz's Maple program *)

a(n+1) is Weigh transform of A005753. - Christian G. Bower, Dec 15 1999
a(n) ~ c * d^n / n^(3/2), where d = A246312 = 5.2490324912281705791649522..., c = 0.05927840588836202377824646... . - Vaclav Kotesovec, Aug 25 2014
G.f. A(x) satisfies: A(x) = x * exp( A(x)^2/x - A(x^2)^2/(2*x^2) + A(x^3)^2/(3*x^3) - A(x^4)^2/(4*x^4) + ... ). - Ilya Gutkovskiy, May 26 2023

More terms from Christian G. Bower, Dec 15 1999

A245870 Decimal expansion of a constant related to A000151.

Original entry on oeis.org

5, 6, 4, 6, 5, 4, 2, 6, 1, 6, 2, 3, 2, 9, 4, 9, 7, 1, 2, 8, 9, 2, 7, 1, 3, 5, 1, 6, 2, 1, 6, 9, 1, 3, 8, 3, 8, 1, 4, 9, 8, 2, 1, 9, 1, 1, 6, 0, 5, 3, 8, 4, 3, 9, 2, 3, 8, 5, 8, 1, 7, 0, 2, 8, 8, 5, 0, 0, 2, 1, 4, 3, 1, 1, 2, 2, 4, 9, 4, 3, 0, 7, 7, 0, 7, 4, 2, 7, 5, 5, 5, 1, 6, 1, 1, 7, 7, 8, 8, 3, 4, 0, 6, 6, 0

Offset: 1

Vaclav Kotesovec, Aug 25 2014

			5.646542616232949712892713516216913838149821911605384392385817...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 307 and 564.

Cf. A000151, A000238, A005750, A005751, A038055, A198760.

Equals lim n -> infinity A000151(n)^(1/n).
Equals lim n -> infinity A005751(n)^(1/n).
Equals lim n -> infinity A038055(n)^(1/n).
Equals lim n -> infinity A005750(n)^(1/n).
Equals lim n -> infinity A198760(n)^(1/n).

More terms from Vaclav Kotesovec, Sep 11 2014 and Dec 26 2020

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

A363387 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^2 / (k*x^k) ).

Original entry on oeis.org

1, 1, 1, 3, 6, 17, 42, 120, 330, 962, 2797, 8334, 24989, 75905, 232142, 715830, 2220473, 6928411, 21723883, 68424327, 216376757, 686742855, 2186771571, 6984248840, 22368127861, 71818903891, 231132440916, 745454242656, 2409080380316, 7799945417349

Offset: 1

Ilya Gutkovskiy, May 30 2023

nonn

Cf. A005750, A007562, A363388.

nmax = 30; A[] = 0; Do[A[x] = x + x^2 Exp[Sum[A[x^k]^2/(k x^k), {k, 1, nmax}]] + O[x]^(nmax + 1)//Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; g[n_] := g[n] = Sum[a[k] a[n - k], {k, 1, n - 1}]; a[n_] := a[n] = (1/(n - 2)) Sum[Sum[d g[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 30}]

seq(n)=my(p=x+x^2+O(x^3)); for(n=1, n\2, my(m=serprec(p,x)-1); p = x + x^2*exp(sum(k=1, m, subst(p + O(x^(m\k+1)), x, x^k)^2/(x^k*k)))); Vec(p + O(x*x^n)) \\ Andrew Howroyd, May 30 2023

A005755 Number of identity matched trees with n nodes.

Original entry on oeis.org

0, 0, 0, 1, 4, 16, 64, 252, 1018, 4182, 17510, 74510, 322034, 1410362, 6251114, 27998532, 126583634, 577079333, 2650573354, 12256481666, 57021299394, 266754944481, 1254245360430, 5924659521632, 28105641930102, 133853504339029, 639801068848128

Offset: 1

N. J. A. Sloane

nonn

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..450
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. A005753, A005754, A102755, A246312.

with(numtheory): b2:= proc(n) option remember; local m; `if`(n=1, 1, 2/(n-1) *add(b2(m) *add((-1)^((n-m)/d+1) *d*b2(d), d=divisors(n-m)), m=1..n-1)) end: c2:= proc(n) option remember; local m; `if`(n=1, 1, 1/(n-1) *add(c2(m) *add((-1)^((n-m)/d+1) *d*b2(d), d=divisors(n-m)), m=1..n-1)) end: a2:= n-> (b2(n) -add(b2(m) *b2(n-m), m=1..n-1) -`if`(irem(n, 2)=0, b2(n/2), c2((n+1)/2)))/2: seq(a2(n), n=1..30); # Alois P. Heinz, Aug 04 2009

b2[n_] := b2[n] = If [n == 1, 1, 2/(n-1)*Sum[b2[m]*Sum[(-1)^((n-m)/d+1)*d*b2[d], {d, Divisors[n-m]}], {m, 1, n-1}]]; c2[n_] := c2[n] = If [n == 1, 1, 1/(n-1)*Sum[c2[m]*Sum[(-1)^((n-m)/d+1)*d*b2[d], {d, Divisors[n-m]}], {m, 1, n-1}]]; a2[n_] := (b2[n] - Sum[b2[m]*b2[n-m], {m, 1, n-1}] - If[Mod[n, 2] == 0, b2[n/2], c2[(n+1)/2]])/2; Table[a2[n], {n, 1, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

a(n) ~ c * d^n / n^(5/2), where d = A246312 = 5.2490324912281705791649522..., c = 0.089035519570392123219315... . - Vaclav Kotesovec, Aug 25 2014

More terms from Alois P. Heinz, Aug 04 2009

A340814 Array read by antidiagonals: T(n,k) is the number of unlabeled oriented edge-rooted k-gonal 2-trees with n oriented polygons, n >= 0, k >= 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 9, 1, 1, 5, 19, 39, 20, 1, 1, 6, 31, 107, 160, 48, 1, 1, 7, 46, 229, 647, 702, 115, 1, 1, 8, 64, 421, 1832, 4167, 3177, 286, 1, 1, 9, 85, 699, 4191, 15583, 27847, 14830, 719, 1, 1, 10, 109, 1079, 8325, 44322, 137791, 191747, 70678, 1842

Offset: 0

Andrew Howroyd, Feb 02 2021

See section 2 of the Labelle reference.

			Array begins:
============================================================
n\k |   2     3      4       5       6        7        8
----+-------------------------------------------------------
  0 |   1     1      1       1       1        1        1 ...
  1 |   1     1      1       1       1        1        1 ...
  2 |   2     3      4       5       6        7        8 ...
  3 |   4    10     19      31      46       64       85 ...
  4 |   9    39    107     229     421      699     1079 ...
  5 |  20   160    647    1832    4191     8325    14960 ...
  6 |  48   702   4167   15583   44322   105284   220193 ...
  7 | 115  3177  27847  137791  487662  1385888  3374267 ...
  8 | 286 14830 191747 1255202 5527722 18795035 53275581 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. Labelle, C. Lamathe and P. Leroux, Labeled and unlabeled enumeration of k-gonal 2-trees, arXiv:math/0312424 [math.CO], Dec 23 2003.

Columns k=2..6 are A000081(n+1), A005750(n+1), A052751, A052773, A052781.

Cf. A242249, A340811, A340812.

\\ here B(n,k) gives g.f. of k-th column.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(n, k)={my(p=1+O(x)); for(n=1, n, p=1+x*Ser(EulerT(Vec(p^(k-1))))); p}
{ Mat(vector(7, k, Col(B(7, k+1)))) }

Column k is the Euler transform of column k+1 of A242249.

G.f. of column k: A(x) satisfies A(x) = exp(Sum_{i>0} x^i*A(x^i)^(k-1)/i).

A005752 a(n) = n^2 + nfloor(ntau) - floor(n*tau)^2.

Original entry on oeis.org

0, 1, 1, 5, 4, 1, 9, 5, 16, 11, 4, 19, 11, 1, 20, 9, 31, 19, 5, 31, 16, 45, 29, 11, 44, 25, 4, 41, 19, 59, 36, 11, 55, 29, 1, 49, 20, 71, 41, 9, 64, 31, 89, 55, 19, 81, 44, 5, 71, 31, 100, 59, 16, 89, 45, 121, 76, 29, 109

Offset: 0

N. J. A. Sloane

nonn

P. Fahr, Infinite Gabriel-Roiter measures for the 3-Kronecker quiver, PhD Thesis U. Bielefeld (2008) p 45
Clark Kimberling, The equation m^2 - 4k = 5n^2 and unique representations of positive integers, Fibonacci Quart. 45 (2007), no. 4, 304-312.
Rodica Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97.
Rodica Simion, Trees with 1-factors and oriented trees, Discrete Math., 88 (1981), 97. (Annotated scanned copy)

Table[n^2 + n Floor[n #] - Floor[n #]^2 &@ GoldenRatio, {n, 0, 60}] (* Michael De Vlieger, Mar 06 2016 *)

a(n) = my(fnt = floor(n*(sqrt(5)+1)/2));  n^2 + n*fnt - fnt^2; \\ Michel Marcus, Mar 05 2016

A007748 Number of self-converse oriented trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 10, 26, 39, 107, 160, 458, 702, 2058, 3177, 9498, 14830, 44947, 70678, 216598, 342860, 1059952, 1686486, 5251806, 8393681, 26297238, 42187148, 132856766, 213828802, 676398395, 1091711076

Offset: 1

N. J. A. Sloane

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)
Index entries for sequences related to trees

Cf. A000238.

max = 15; 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)]; a[n_] := A[n, 2]; A000151 = Table[a[n], {n, 1, max}]; etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; A005750 = Table[etr[a][n], {n, 0, max}] ; A007748 = Riffle[A005750, A000151] (* Jean-François Alcover, Jul 16 2015 *)

a(2n)=A000151(n). a(2n-1)=A005750(n). - Christian G. Bower, Dec 15 1999

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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A005753 Number of rooted identity matched trees with n nodes.

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A005754 Number of planted identity matched trees with n nodes.

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A245870 Decimal expansion of a constant related to A000151.

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A005751 Number of matched trees with 2n nodes.

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A363387 G.f. A(x) satisfies: A(x) = x + x^2 * exp( Sum_{k>=1} A(x^k)^2 / (k*x^k) ).

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A005755 Number of identity matched trees with n nodes.

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A005752 a(n) = n^2 + nfloor(ntau) - floor(n*tau)^2.