A000248 -id:A000248 - cp's OEIS frontend

A052512 Number of rooted labeled trees of height at most 2.

0, 1, 2, 9, 40, 205, 1176, 7399, 50576, 372537, 2936080, 24617131, 218521128, 2045278261, 20112821288, 207162957135, 2228888801056, 24989309310961, 291322555295904, 3524580202643155, 44176839081266360, 572725044269255661, 7668896804574138232, 105920137923940473079

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Equivalently, number of mappings f from a set of n elements into itself such that f o f (f applied twice) is constant. - Robert FERREOL, Mar 05 2016

			From _Robert FERREOL_, Mar 05 2016: (Start)
For n = 3 the a(3) = 9 mappings from {a,b,c} into itself are:
f_1(a) = f_1(b) = f_1(c) = a
f_2(c) = b, f_2(b) = f_2(a) = a
f_3(b) = c, f_3(c) = f_3(a) = a
and 6 others, associated to b and c.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 58

Cf. A000248 (forests with n nodes and height at most 1).

Cf. A000551.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( x*Exp(x*Exp(x)) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, May 13 2019

spec := [S,{S=Prod(Z,Set(T1)), T2=Z, T1=Prod(Z,Set(T2))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
a:= n-> n*add(binomial(n-1, k)*(n-k-1)^k, k=0..n-1);
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 15 2013

nn=20; a=x Exp[x]; Range[0,nn]! CoefficientList[Series[x Exp[a], {x,0,nn}], x] (* Geoffrey Critzer, Sep 19 2012 *)

N=33;  x='x+O('x^N);
egf=x*exp(x*exp(x));
v=Vec(serlaplace(egf));
vector(#v+1,n,if(n==1,0,v[n-1]))
/* Joerg Arndt, Sep 15 2012 */

m = 20; T = taylor(x*exp(x*exp(x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019

E.g.f.: x*exp(x*exp(x)).
a(n) = n * A000248(n-1). - Olivier Gérard, Aug 03 2012.
a(n) = Sum_{k=0..n-1} n*C(n-1,k)*(n-k-1)^k. - Alois P. Heinz, Mar 15 2013

A007550 Natural numbers exponentiated twice.

Original entry on oeis.org

1, 4, 20, 127, 967, 8549, 85829, 962308, 11895252, 160475855, 2343491207, 36795832297, 617662302441, 11031160457672, 208736299803440, 4169680371133507, 87648971646028515, 1933298000313801349, 44633323736412392093, 1076069422794010119112

Offset: 1

N. J. A. Sloane

The subsequence of primes (for n = 4, 5, 7) begins: 127, 967, 85829. The subsequence of semiprimes (for n = 2, 6) begins: 4, 8549. - Jonathan Vos Post, Feb 09 2011

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..200
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 166
N. J. A. Sloane, Transforms

exptr:= proc(p) local g; g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: end: a:= exptr(exptr(n->n)): seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008

a[n_] := Sum[k^(n-k)*Binomial[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 11 2014, after Olivier Gérard *)

E.g.f.: exp(G(x) - 1) - 1, where G(x) = exp(x*exp(x)) = e.g.f. for A000248; clarified by Ilya Gutkovskiy, Jun 25 2018

a(n) = sum( k^(n - k) binomial(n,k) bell(k), k = 0..n ). - Olivier Gérard, Oct 24 2007

A135746 E.g.f.: A(x) = Sum_{n>=0} exp(n^2x) x^n/n!.

Original entry on oeis.org

1, 1, 3, 16, 137, 1536, 22417, 407884, 8920641, 230576320, 6928080641, 238375169484, 9288784476193, 406150114297552, 19761959813464065, 1062437048084297596, 62727815353861478273, 4045278841893314992896

Offset: 0

Paul D. Hanna, Nov 27 2007

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 137*x^4/4! + 1536*x^5/5! + ...
where A(x) = 1 + exp(x)*x + exp(4*x)*x^2/2! + exp(9*x)*x^3/3! + exp(16*x)*x^4/4! + exp(25*x)*x^5/5! + ...
O.g.f.: F(x) = 1 + x + 3*x^2 + 16*x^3 + 137*x^4 + 1536*x^5 + 22417*x^6 + ...
where F(x) = 1 + x/(1-x)^2 + x^2/(1-4*x)^3 + x^3/(1-9*x)^4 + x^4/(1-16*x)^5 + x^5/(1-25*x)^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. variants: A135742, A135743, A135744, A135745, A135747.

Cf. A000248.

Flatten[{1, Table[Sum[Binomial[n, k]*k^(2*(n - k)), {k, 0, n}], {n, 1, 25}]}] (* G. C. Greubel, Nov 05 2016 *)

{a(n)=sum(k=0,n,binomial(n,k)*(k^2)^(n-k))}

{a(n)=n!*polcoeff(sum(k=0,n,exp(k^2*x +x*O(x^n))*x^k/k!),n)}

{a(n)=polcoeff(sum(k=0, n, x^k/(1-k^2*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Aug 08 2009

a(n) = Sum_{k=0..n} C(n,k)*(k^2)^(n-k).
O.g.f.: Sum_{n>=0} x^n/(1 - n^2*x)^(n+1). - Paul D. Hanna, Aug 08 2009
a(n) ~ n^(n + 1/2) * r^(2*n - 3*r + 1/2) / (sqrt(2*n + 3*r) * (n - r)^(n - r + 1/2)), where r = (n/w) * (1 + (w-1)/((2*w^2 + w - 2)/log(w-1) - w + 2)) and w = LambertW(exp(1)*n). - Vaclav Kotesovec, Jul 05 2022

A177208 Numerators of exponential transform of 1/n.

Original entry on oeis.org

1, 1, 3, 17, 19, 81, 8351, 184553, 52907, 1768847, 70442753, 1096172081, 22198464713, 195894185831, 42653714271997, 30188596935106763, 20689743895700791, 670597992748852241, 71867806446352961329, 8445943795439038164379, 379371134635840861537

Offset: 0

Franklin T. Adams-Watters, May 04 2010

b(n) = a(n)/A177209(n) is the sum over all set partitions of [n] of the product of the reciprocals of the part sizes.

Numerators of moments of Dickman-De Bruijn distribution as shown on page 257 of Cellarosi and Sinai. [Jonathan Vos Post, Jan 07 2012]

			For n=4, there is 1 set partition with a single part of size 4, 4 with sizes [3,1], 3 with sizes [2,2], 6 with sizes [2,1,1], and 1 with sizes [1,1,1,1]; so b(4) = 1/4 + 4/(3*1) + 3/(2*2) + 6/(2*1*1) + 1/(1^4) = 1/4 + 4/3 + 3/4 + 3 + 1 = 19/4.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), pp. 228-230.
Knuth, Donald E., and Luis Trabb Pardo. "Analysis of a simple factorization algorithm." Theoretical Computer Science 3.3 (1976): 321-348. See Eq. (6.6) and (6.7), page 334.

Alois P. Heinz, Table of n, a(n) for n = 0..200
F. Cellarosi and Ya. G. Sinai, The Möbius function and statistical mechanics, Bull. Math. Sci., 2011.
Wikipedia, Exponential integral

Denominators are in A177209.

Cf. A000110, A000248, A001620, A322364, A322365, A322380, A322381, A323339, A323340.

b:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*b(n-j)/j, j=1..n))
    end:
a:= n-> numer(b(n)):
seq(a(n), n=0..25); # Alois P. Heinz, Jan 08 2012

b[n_] := b[n] = If[n==0, 1, Sum[Binomial[n-1, j-1]*b[n-j]/j, {j, 1, n}]]; a[n_] := Numerator[b[n]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)

Vec(serlaplace(exp(sum(n=1,30,x^n/(n*n!),O(x^31)))))

E.g.f. for fractions is exp(f(z)), where f(z) = sum(k>0, z^k/(k*k!)) = integral(0..z,(exp(t)-1)/t dt) = Ei(z) - gamma - log(z) = -Ein(-z). Here gamma is Euler's constant, and Ei and Ein are variants of the exponential integral.

Knuth & Trabb-Pardo (6.7) gives a recurrence. - N. J. A. Sloane, Nov 09 2022

A235328 Number of ordered pairs of endofunctions (f,g) on a set of n elements satisfying f(x) = g(f(f(x))).

Original entry on oeis.org

1, 1, 6, 69, 1336, 39145, 1598256, 85996561, 5872177536, 494848403793, 50333180780800, 6068500612311841, 854434117410352128, 138752719761249646585, 25714777079368557164544, 5389541081414619785888625, 1267387594395443339970052096, 332074775201035547446532113825

Offset: 0

Chad Brewbaker, Mar 26 2014

nonn

This also counts pairs (f,g) satisfying f(x) = g(f^{r}(x)) for r > 1. - David Einstein, Nov 18 2016

Alois P. Heinz, Table of n, a(n) for n = 0..263

Cf. A000248, A000949, A181162, A351795.

a:= proc(n) option remember; local b; b:=
      proc(m, i) option remember; `if`(m=0, n^i, `if`(i<1, 0,
        add(b(m-j, i-1)*binomial(m, j)*j, j=0..m)))
      end: forget(b):
      b(n$2)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 23 2014

a[n_] := If[n==0, 1, Sum[k! Binomial[n, k] (n k)^(n - k), {k, 1, n}]]
  Table[a[n],{n,20}] (* David Einstein, Oct 10 2016 *)

a(n) = Sum_{k=1..n} k! * C(n, k) * (n*k)^(n-k). - David Einstein, Oct 10 2016
a(n) = n! * [x^n] 1/(1 - x*exp(n*x)). - Ilya Gutkovskiy, Nov 26 2017
log(a(n)) ~ log(sqrt(2*Pi) * n^(2*n - n/LambertW(exp(1)*n) + 1/2) / (LambertW(exp(1)*n) * exp(n/LambertW(exp(1)*n)) * (LambertW(exp(1)*n) - 1)^(n*(1 - 1/LambertW(exp(1)*n))))). - Vaclav Kotesovec, Feb 20 2022
More precise asymptotics: a(n) ~ sqrt(2*Pi) * (w^2 - w - 1 + 2/w) * exp(n*(1/w^3 - 1/w)) * n^(2*n + n/w^3 - n/w + 1/2) * (w^2 - 1)^(n*(1 + 1/w^3 - 1/w)) * (1 - w^2 + w^3)^(n/w - n - n/w^3 - 1), where w = LambertW(exp(1)*n). - Vaclav Kotesovec, Feb 23 2022

a(6)-a(7) from Giovanni Resta, Mar 26 2014

a(8)-a(17) from Alois P. Heinz, Jul 23 2014

A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 390, 2562, 11816, 105912, 1063530, 8815070, 81342492, 895185876, 9971185406, 112642410090, 1372455608400, 17750397057392, 236950003516626, 3286258330135734, 47688868443593540, 719345273005797900, 11222288509573985382, 181168865439054099266

Offset: 0

Vaclav Kotesovec, Aug 06 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A292892.
Cf. A000110, A052506, A216507, A060311.
Cf. A216688, A216689, A080108, A000248.

CoefficientList[Series[E^(x^2*(E^x-1)), {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^30); Vec(serlaplace(exp(x^2*(exp(x) - 1)))) \\ G. C. Greubel, Nov 21 2017

a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

a(n) ~ exp((n-r^3)/(2+r)-n) * n^(n+1/2) / (r^n * sqrt((2*r^3*(3+r) + n*(1+r)*(4+r))/(2+r))), where r is the root of the equation r^2*((2+r) * exp(r) - 2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(n-2*k)!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)

A009121 Expansion of e.g.f. cosh(exp(x)*x).

Original entry on oeis.org

1, 0, 1, 6, 25, 100, 481, 2954, 20721, 151848, 1146721, 9111982, 77652169, 710421452, 6891125697, 69961213170, 738718169569, 8108554524112, 92647353941569, 1101958783026134, 13616813607795321, 174287243264606388, 2304515271134124577, 31424734896799742170

Offset: 0

R. H. Hardin

Iain Fox, Table of n, a(n) for n = 0..541

Cf. A000248, A009017, A009448, A009565, A009635, A009768.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Cosh(x*Exp(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jul 26 2018

a:= n-> add(`if`(k::odd, 0, binomial(n, k)*k^(n-k)), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 15 2018

With[{nn=30},CoefficientList[Series[Cosh[Exp[x]*x],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 28 2015 *)

first(n) = x='x+O('x^n); Vec(serlaplace(cosh(exp(x)*x))) \\ Iain Fox, Dec 23 2017

Extended and signs tested by Olivier Gérard, Mar 15 1997

Definition clarified and prior Mathematica program replaced by Harvey P. Dale, Dec 28 2015

A059300 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 4.

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 12, 24, 4, 1, 20, 90, 80, 5, 1, 30, 240, 540, 240, 6, 1, 42, 525, 2240, 2835, 672, 7, 1, 56, 1008, 7000, 17920, 13608, 1792, 8, 1, 72, 1764, 18144, 78750, 129024, 61236, 4608, 9, 1, 90, 2880, 41160, 272160, 787500, 860160, 262440, 11520, 10

Offset: 0

N. J. A. Sloane, Jan 25 2001

			Triangle begins:
1;
1,  2;
1,  6,   3;
1, 12,  24,    4;
1, 20,  90,   80,    5;
1, 30, 240,  540,  240,   6;
1, 42, 525, 2240, 2835, 672, 7;
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
John Riordan and N. J. A. Sloane, Correspondence, 1974

There are 4 versions: A059297-A059300. Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc. Row sums are A000248.

/* As triangle: */ [[Binomial(n+1,n-k+1)*(n-k+1)^k: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

t[n_, k_] := Binomial[n + 1, k]*(n - k + 1)^k; Flatten@Table[t[n, k], {n, 0, 9}, {k, 0, n}] (* Arkadiusz Wesolowski, Mar 23 2013 *)

for(n=0, 25, for(k=0, n, print1(binomial(n+1,k)*(n-k+1)^k, ", "))) \\ G. C. Greubel, Jan 05 2017

T(n,k) = binomial(n+1,n-k+1)*(n-k+1)^k. - R. J. Mathar, Mar 14 2013

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

A354553 Expansion of e.g.f. exp( x * exp(x^3) ).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 3361, 42001, 275185, 1819441, 30777121, 371238121, 3057311401, 44263763545, 801096528961, 9710981323681, 125367419194081, 2643123767954401, 45840730383002305, 646414025466298681, 13258301279836276441

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..475

Cf. A000248, A216688, A354554.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^3)))))

a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(k!*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(k! * (n - 3*k)!).

cp's OEIS Frontend

A052512 Number of rooted labeled trees of height at most 2.

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A007550 Natural numbers exponentiated twice.

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A135746 E.g.f.: A(x) = Sum_{n>=0} exp(n^2*x) * x^n/n!.

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A177208 Numerators of exponential transform of 1/n.

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A235328 Number of ordered pairs of endofunctions (f,g) on a set of n elements satisfying f(x) = g(f(f(x))).

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A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

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A009121 Expansion of e.g.f. cosh(exp(x)*x).

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A135746 E.g.f.: A(x) = Sum_{n>=0} exp(n^2x) x^n/n!.