A102755 -id:A102755 - cp's OEIS frontend

A005754 Number of planted identity matched trees with n nodes.

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

A246312 Decimal expansion of a constant related to identity matched trees.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Aug 25 2014

			5.249032491228170579164952216184309265343086337648736503202233186059585565...

Cf. A005753, A005754, A005755, A102755, A038078, A038077.

Equals lim n -> infinity A005753(n)^(1/n).
Equals lim n -> infinity A005754(n)^(1/n).
Equals lim n -> infinity A005755(n)^(1/n).
Equals lim n -> infinity A102755(n)^(1/n).
Equals lim n -> infinity A038078(n)^(1/n).

More terms from Vaclav Kotesovec, Sep 06 2014, Feb 24 2015 and Dec 26 2020

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

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A005754 Number of planted identity matched trees with n nodes.

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A246312 Decimal expansion of a constant related to identity matched trees.

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A005755 Number of identity matched trees with n nodes.

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