A000951 -id:A000951 - cp's OEIS frontend

A181504 A000272(n+1) - A000951(n), n >= 6. Number of forests of rooted labeled trees with n nodes and height >= 5.

720, 32760, 1095360, 33974640, 1054196640, 33765565680, 1132318630080, 40012680759480, 1493837702076720, 58961961488790360, 2459382489787466880, 108300904378742056800, 5028206935516237005120

Offset: 6

Washington Bomfim, Oct 24 2010

nonn

Cf. A000272, A000951.

a(n) = A000272(n+1) - A000951(n), n >= 6.

A000949 Number of forests with n nodes and height at most 2.

Original entry on oeis.org

1, 1, 3, 16, 101, 756, 6607, 65794, 733833, 9046648, 121961051, 1782690174, 28055070397, 472594822324, 8479144213191, 161340195463066, 3243707386310033, 68679247688467056, 1526976223741111987, 35557878951515668726, 865217354118762606021

Offset: 0

N. J. A. Sloane

nonn

Equivalently, the number of mappings from a set of n elements into itself where f(f(x)) = f(f(f(x))). - Chad Brewbaker, Mar 26 2014

			G.f. = 1 + x + 3*x^2 + 16*x^3 + 101*x^4 + 756*x^5 + 6607*x^6 + 65794*x^7 + ... - _Michael Somos_, Jul 03 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 = 0..100
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Cf. A000248, A000950, A000951, A052512-A052514.

Column k=2 of A210725. - Alois P. Heinz, Mar 15 2013

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)
a[ n_] := If[ n < 0, 0, 1 + n! Sum[ Sum[ k^(n - m - k) m^k / (k! (n - m - k)!), {k, n - m}] / m!, {m, n - 1}]]; (* Michael Somos, Jul 03 2018 *)

a(n):=n!*sum(sum((k^(n-m-k)*m^k)/(k!*(n-m-k)!),k,1,n-m)/m!,m,1,n-1)+1; /* Vladimir Kruchinin, May 28 2011 */

x='x+O('x^66); Vec(serlaplace(exp(x*exp(x*exp(x))))) /* show terms with a(0)=1 */ /* Joerg Arndt, May 28 2011 */

E.g.f.: exp(x*exp(x*exp(x))).

a(n) = n!*sum(m=1..n-1, sum(k=1..n-m, (k^(n-m-k)*m^k)/(k!*(n-m-k)!))/m!)+1. - Vladimir Kruchinin, May 28 2011

More terms from Vladeta Jovovic, Apr 07 2001

A060905 Expansion of e.g.f. exp(xexp(x) + 1/2x^2*exp(x)^2).

Original entry on oeis.org

1, 1, 4, 19, 110, 751, 5902, 52165, 509588, 5437729, 62828306, 780287839, 10351912276, 145944541159, 2176931651546, 34225419288421, 565282627986368, 9779830102138945, 176776613812205074, 3330780287838743575

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

Number of functions f from a set of size n to itself such that f(f(f(x))) = f(x). - Joel B. Lewis, Dec 12 2011

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A000949 - A000951, A060905 - A060913.

Column k=3 of A245501.

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^2/2],{x,0,nn}],x]  (* Geoffrey Critzer, Sep 18 2012 *)

a(n):=sum(sum(k^(n-k)/(n-k)!*binomial(m,k-m)*(1/2)^(k-m),k,m,n)/m!,m,1,n); /* Vladimir Kruchinin, Aug 20 2010 */

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 2.

a(n) = sum(sum(k^(n-k)/(n-k)!*binomial(m,k-m)*(1/2)^(k-m),k,m,n)/m!,m,1,n), n>0. - Vladimir Kruchinin, Aug 20 2010

A060913 E.g.f.: exp(xexp(xexp(xexp(x))) + 1/3x^3exp(xexp(x*exp(x)))^3).

Original entry on oeis.org

1, 1, 3, 18, 157, 1656, 20727, 300784, 4955337, 91229616, 1853584651, 41147256624, 989990665677, 25647894553048, 711630284942319, 21049888453838136, 661180220075555473, 21976354057916680416

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

a(n) is the number of functions f:{1,2,...,n} -> {1,2,...,n} such that the functional digraphs have cycles of length 1 or 3 and no element is at a distance of more than 3 from a cycle. - Geoffrey Critzer, Sep 23 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A000949-A000951, A060905-A060913.

nn=20; a=x Exp[x]; b=x Exp[a]; c=x Exp[b]; t=Sum[n^(n-1)x^n/n!, {n, 1, nn}]; Range[0,nn]! CoefficientList[Series[Exp[c+c^3/3], {x, 0, nn}], x] (* Geoffrey Critzer, Sep 23 2012 *)

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 3, m = 3.

A210725 Triangle read by rows: T(n,k) = number of forests of labeled rooted trees with n nodes and height at most k (n>=1, 0<=k<=n-1).

Original entry on oeis.org

1, 1, 3, 1, 10, 16, 1, 41, 101, 125, 1, 196, 756, 1176, 1296, 1, 1057, 6607, 12847, 16087, 16807, 1, 6322, 65794, 160504, 229384, 257104, 262144, 1, 41393, 733833, 2261289, 3687609, 4480569, 4742649, 4782969, 1, 293608, 9046648, 35464816, 66025360, 87238720, 96915520, 99637120, 100000000

Offset: 1

N. J. A. Sloane, May 09 2012

			Triangle begins:
  1;
  1,    3;
  1,   10,   16;
  1,   41,  101,   125;
  1,  196,  756,  1176,  1296;
  1, 1057, 6607, 12847, 16087, 16807;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Diagonals include A000248, A000949, A000950, A000951, A000272.

f:= proc(k) f(k):= `if`(k<0, 1, exp(x*f(k-1))) end:
T:= (n, k)-> coeff(series(f(k), x, n+1), x, n) *n!:
seq(seq(T(n, k), k=0..n-1), n=1..9); # Alois P. Heinz, May 30 2012
# second Maple program:
T:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
      binomial(n-1, j-1)*j*T(j-1, h-1)*T(n-j, h), j=1..n))
    end:
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Aug 21 2017

f[?Negative] = 1; f[k] := Exp[x*f[k-1]]; t[n_, k_] := Coefficient[Series[f[k], {x, 0, n+1}], x, n]*n!; Table[Table[t[n, k], {k, 0, n-1}], {n, 1, 9}] // Flatten (* Jean-François Alcover, Oct 30 2013, after Maple *)

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def T(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*T(j - 1, h - 1)*T(n - j, h) for j in range(1, n + 1)])
for n in range(1, 11): print([T(n, k) for k in range(n)]) # Indranil Ghosh, Aug 21 2017, after second Maple code

A000950 Number of forests with n nodes and height at most 3.

Original entry on oeis.org

1, 3, 16, 125, 1176, 12847, 160504, 2261289, 35464816, 612419291, 11539360944, 235469524237, 5170808565976, 121535533284999, 3043254281853496, 80852247370051793, 2270951670959226336, 67221368736302224819, 2091039845329887687136

Offset: 1

N. J. A. Sloane

nonn

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..100
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Cf. A000248, A000949-A000951, A052512-A052514.

Column k=3 of A210725. - Alois P. Heinz, Mar 15 2013

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[x*Exp[x*Exp[x*Exp[x]]]], {x, 0, nn}], x] (* T. D. Noe, Jun 21 2012 *)

E.g.f.: exp(x*exp(x*exp(x*exp(x)))).

More terms from Vladeta Jovovic, Apr 07 2001

A060906 E.g.f.: exp(xexp(x) + 1/3x^3*exp(x)^3).

Original entry on oeis.org

1, 1, 3, 12, 73, 556, 4737, 44122, 453441, 5186664, 65671201, 906052654, 13418086497, 211472682604, 3535616946513, 62621439810066, 1172370604136833, 23118679430573008, 478329265510033473, 10349724555927678934, 233633352312272612001, 5492655756487132979796

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

The number of functions from {1,2,...,n} to itself such that f(x)=f^4(x). - Geoffrey Critzer, Sep 18 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000949 - A000951, A060905 - A060913.

Column k=4 of A245501.

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^3/3],{x,0,nn}],x] (* Geoffrey Critzer, Sep 18 2012 *)

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 3.

A060907 E.g.f.: exp(xexp(x) + 1/2x^2exp(x)^2 + 1/4x^4*exp(x)^4).

Original entry on oeis.org

1, 1, 4, 19, 116, 901, 8422, 89755, 1061048, 13746169, 193901066, 2965146559, 48946004956, 867463969789, 16405240966766, 329147315037811, 6973157545554128, 155446026607476145, 3636697161715448914, 89099916704329731895, 2281451214192505136516

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

The number of functions from {1,2,...,n} into itself such that f(x) = f^5(x). - Geoffrey Critzer, Sep 18 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000949 - A000951, A060905 - A060913.

Column k=5 of A245501.

egf:= exp(x*exp(x)+x^2*exp(x)^2/2+x^4*exp(x)^4/4):
a:= n-> n!*coeff(series(egf, x, n+11), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2014

nn=20;a=x Exp[x];Range[0,nn]!CoefficientList[Series[Exp[a+a^2/2+a^4/4],{x,0,nn}],x] (* Geoffrey Critzer, Sep 18 2012 *)

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 1, m = 4.

A060908 E.g.f.: exp(xexp(xexp(x)) + 1/2x^2exp(x*exp(x))^2).

Original entry on oeis.org

1, 1, 4, 25, 194, 1791, 19312, 237637, 3280524, 50136049, 839267936, 15255154179, 298936866736, 6277386102703, 140540145723720, 3339966073612921, 83936496568012208, 2223184658988286113, 61877234830148427808

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

a(n) = the number of functions f:{1,2,...,n} -> {1,2,...,n} such that the functional digraphs have cycle lengths at most 2 and no element is at a distance of more than 2 form a cycle. - Geoffrey Critzer, Sep 23 2012

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A000949-A000951, A060905-A060913.

nn=20; a=x Exp[x]; b=x Exp[a]; t=Sum[n^(n-1)x^n/n! ,{n, 1, nn}]; Range[0,nn]! CoefficientList[Series[Exp[b+b^2/2], {x, 0, nn}], x]  (* Geoffrey Critzer, Sep 23 2012 *)

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 2, m = 2.

A060909 E.g.f.: exp(xexp(xexp(x)) + 1/3x^3exp(x*exp(x))^3).

Original entry on oeis.org

1, 1, 3, 18, 133, 1236, 13767, 176674, 2547561, 40614408, 708601771, 13433957934, 275200324797, 6061423076476, 142868492357151, 3587417860571346, 95560989416582353, 2690066742390963216, 79752454967110250835

Offset: 0

Vladeta Jovovic, Apr 07 2001

nonn

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983.

Cf. A000949 - A000951, A060905 - A060913.

E.g.f.: exp(Sum_{d|m} T_k^d/d), where T_k = x*exp(T_(k - 1)), k >= 1, T_0 = x; k = 2, m = 3.

cp's OEIS Frontend

A181504 A000272(n+1) - A000951(n), n >= 6. Number of forests of rooted labeled trees with n nodes and height >= 5.

Views

Author

Keywords

Crossrefs

Formula

A000949 Number of forests with n nodes and height at most 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

Formula

Extensions

A060905 Expansion of e.g.f. exp(x*exp(x) + 1/2*x^2*exp(x)^2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A060913 E.g.f.: exp(x*exp(x*exp(x*exp(x))) + 1/3*x^3*exp(x*exp(x*exp(x)))^3).

Views

Author

Keywords

Comments

References

Crossrefs

Programs

Mathematica

Formula

A210725 Triangle read by rows: T(n,k) = number of forests of labeled rooted trees with n nodes and height at most k (n>=1, 0<=k<=n-1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A000950 Number of forests with n nodes and height at most 3.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A060906 E.g.f.: exp(x*exp(x) + 1/3*x^3*exp(x)^3).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A060907 E.g.f.: exp(x*exp(x) + 1/2*x^2*exp(x)^2 + 1/4*x^4*exp(x)^4).

Views

Author

A060905 Expansion of e.g.f. exp(xexp(x) + 1/2x^2*exp(x)^2).

A060913 E.g.f.: exp(xexp(xexp(xexp(x))) + 1/3x^3exp(xexp(x*exp(x)))^3).

A060906 E.g.f.: exp(xexp(x) + 1/3x^3*exp(x)^3).

A060907 E.g.f.: exp(xexp(x) + 1/2x^2exp(x)^2 + 1/4x^4*exp(x)^4).

A060908 E.g.f.: exp(xexp(xexp(x)) + 1/2x^2exp(x*exp(x))^2).

A060909 E.g.f.: exp(xexp(xexp(x)) + 1/3x^3exp(x*exp(x))^3).