A072597 -id:A072597 - cp's OEIS frontend

A006153 E.g.f.: 1/(1-x*exp(x)).

1, 1, 4, 21, 148, 1305, 13806, 170401, 2403640, 38143377, 672552730, 13044463641, 276003553860, 6326524990825, 156171026562838, 4130464801497105, 116526877671782896, 3492868475952497313, 110856698175372359346, 3713836169709782989993, 130966414749485504586940

Offset: 0

Simon Plouffe and N. J. A. Sloane

a(n) is the sum of the row entries of triangle A199673, that is, a(n) is the number of ways to assign n people into labeled groups and then to assign a leader for each group from its members; see example below. - Dennis P. Walsh, Nov 15 2011
a(n) is the number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that for some j>1, f^j=f where f^j denotes iterated functional composition. Equivalently, the number of endofunctions such that every element is mapped to a recurrent element. Equivalently, every vertex of the functional digraph is at a distance at most 1 from a cycle. - Geoffrey Critzer, Jan 21 2012
Numerators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." - Michael Somos, Jan 21 2019

			a(3) = 21 since there are 21 ways to assign 3 people into labeled groups with designated leaders. If there is one group, there are 3 ways to select a leader from the 3 people in the group. If there are two groups (group 1 and group 2), there are 6 ways to assign leaders and then 2 ways to select a group for the remaining person, and thus there are 12 assignments. If there are three groups (group1, group 2, and group3), each person is a leader of their singleton group, and there are 6 ways to assign the 3 people to the 3 groups. Hence a(3) = 3 + 12 + 6 = 21.
a(4) = 148 = 4 + 48 + 72 + 24.

S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.32(d).

Alois P. Heinz, Table of n, a(n) for n = 0..200
S. Getu and L. W. Shapiro, Combinatorial view of the composition of functions, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 110
Dennis Walsh, Assigning people into labeled groups with leaders

Row sums of triangle A199673.

Cf. A003566, A072597, A089148.

a := proc(n) local k; add(k^(n-k)*n!/(n-k)!,k=1..n); end; # for n >= 1

With[{nn=20},CoefficientList[Series[1/(1-x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 29 2012 *)
a[ n_] := If[n < 0, 0, n! + n! Sum[(n - k)^k / k!, {k, n}]]; (* Michael Somos, Jan 21 2019 *)

x='x+O('x^66);
egf=1/(1-x*exp(x)); /* = 1 + x + 2*x^2 + 7/2*x^3 + 37/6*x^4 + 87/8*x^5 +... */
Vec(serlaplace(egf)) /* Joerg Arndt, Apr 30 2011 */

{a(n) = if(n<0, 0, n! * sum(k=0, n, (n-k)^k / k!))}; /* Michael Somos, Jan 21 2019 */

def A006153_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        for k in range(n, 0, -1):
            C[k] = -C[k-1]*(1/(k-1) if k>1 else 1)
        C[0] = sum((-1)^k*C[k] for k in (1..n))
        R.append(C[0]*f)
    return R
print(A006153_list(20)) # Peter Luschny, Feb 21 2016

a(n) = n! * Sum_{k=0..n}(n-k)^k/k!.
a(n) = Sum_{k=0..n} k!*k^(n-k)*binomial(n,k).
For n>=1, a(n-1) = b(n) where b(1)=1 and b(n) = Sum_{i=1..n-1} i*binomial(n-1, i)*b(i). - Benoit Cloitre, Nov 13 2004
a(n) = Sum_{k=1..n}A199673(n,k) = Sum_{k=1..n}n! k^(n-k)/(n-k)!. - Dennis P. Walsh, Nov 15 2011
E.g.f. for a(n), n>=1: x*e^x/(1-x*e^x). - Dennis P. Walsh, Nov 15 2011
a(n) ~ n! / ((1+LambertW(1))*LambertW(1)^n). - Vaclav Kotesovec, Jun 21 2013
O.g.f.: Sum_{n>=0} n! * x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018
a(0) = 1; a(n) = n * Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 12 2020

Definition corrected by Joerg Arndt, Apr 30 2011

A152818 Array read by antidiagonals: A(n,k) = (k+1)^n*(n+k)!/n!.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 12, 18, 6, 1, 32, 108, 96, 24, 1, 80, 540, 960, 600, 120, 1, 192, 2430, 7680, 9000, 4320, 720, 1, 448, 10206, 53760, 105000, 90720, 35280, 5040, 1, 1024, 40824, 344064, 1050000, 1451520, 987840, 322560, 40320

Offset: 0

Paul Curtz, Dec 13 2008

A009998/A119502 gives triangle of unreduced coefficients of polynomials defined by A152650/A152656. a(n) gives numerators with denominators n! for each row.

Row 0 is A000142. Row 1 is formed from positive members of A001563. Row 2 is A055533. Column 0 is A000012. Column 1 is formed from positive members of A001787. Column 2 is A006043. Column 3 is A006044. - Omar E. Pol, Jan 06 2009

			From _Omar E. Pol_, Jan 06 2009: (Start)
Array begins:
  1,    1,      2,        6,         24,          120, ...
  1,    4,     18,       96,        600,         4320, ...
  1,   12,    108,      960,       9000,        90720, ...
  1,   32,    540,     7680,     105000,      1451520, ...
  1,   80,   2430,    53760,    1050000,     19595520, ...
  1,  192,  10206,   344064,    9450000,    235146240, ...
  1,  448,  40824,  2064384,   78750000,   2586608640, ...
  1, 1024, 157464, 11796480,  618750000,  26605117440, ...
  1, 2304, 590490, 64880640, 4640625000, 259399895040, ... (End)
Antidiagonal triangle:
  1;
  1,   1;
  1,   4,     2;
  1,  12,    18,     6;
  1,  32,   108,    96,     24;
  1,  80,   540,   960,    600,   120;
  1, 192,  2430,  7680,   9000,  4320,   720;
  1, 448, 10206, 53760, 105000, 90720, 35280, 5040;

G. C. Greubel, Antidiagonals n = 0..50, flattened
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. See page 422.
F. A. Haight, Overflow at a traffic light, Biometrika, 46 (1959), 420-424. (Annotated scanned copy)
F. A. Haight, Letter to N. J. A. Sloane, n.d.

Cf. A000012, A000142, A001563, A001787, A006043, A006044, A009998.
Cf. A055533, A072597 (antidiagonal sums), A089148, A119502, A152650.
Cf. A152656, A154372.

A152818:= func< n,k | (k+1)^(n-k)*Factorial(k)*Binomial(n,k) >;
[A152818(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 10 2023

len= 45; m= 1 + Ceiling[Sqrt[len]]; Sort[Flatten[#, 1] &[MapIndexed[ {(2 +#2[[1]]^2 +(#2[[2]] -1)*#2[[2]] +#2[[1]]*(2*#2[[2]] -3))/ 2, #1}&, Table[(k+1)^n*(n+k)!/n!, {n,0,m}, {k,0,m}], {2}]]][[All, 2]][[1 ;; len]] (* From Jean-François Alcover, May 27 2011 *)
T[n_, k_]:= (k+1)^(n-k)*k!*Binomial[n, k];
Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 10 2023 *)

A(n,k) = (k+1)^n*(n+k)!/n! \\ Charles R Greathouse IV, Sep 10 2016

def A152818_row(n):
    R. = ZZ[]
    P = add((n-k+1)^k*x^(n-k+1)*factorial(n)/factorial(k) for k in (0..n))
    return P.coefficients()
for n in (0..12): print(A152818_row(n))  # Peter Luschny, May 03 2013

E.g.f. for array as a triangle: exp(x)/(1-t*x*exp(x)) = 1+(1+t)*x+(1+4*t+2*t^2)*x^2/2! + (1+12*t+18*t^2+6*t^3)*x^3/3! + .... E.g.f. is int {z = 0..inf} exp(-z)*F(x,t*z), (x and t chosen sufficiently small for the integral to converge), where F(x,t) = exp(x*(1+t*exp(x))) is the e.g.f. for A154372. - Peter Bala, Oct 09 2011
From Peter Bala, Oct 09 2011: (Start)
From the e.g.f., the row polynomials R(n,t) satisfy the recursion R(n,t) = 1 + t*sum {k = 0..n-1} n!/(k!*(n-k-1)!)*R(n-k-1,t). The polynomials 1/n!*R(n,x) are the polynomials P(n,x) of A152650.
Sum_{k=0..n} T(n, k) = A072597(n) (antidiagonal sums). (End)
From G. C. Greubel, Apr 10 2023: (Start)
T(n, k) = (k+1)^(n-k) * k! * binomial(n, k) (antidiagonal triangle).
Sum_{k=0..n} (-1)^k*T(n, k) = A089148(n). (End)

Better definition, extended and edited by Omar E. Pol and N. J. A. Sloane, Jan 05 2009

A305990 Expansion of e.g.f.: (1+x) / (exp(-x) - x).

Original entry on oeis.org

1, 3, 11, 58, 409, 3606, 38149, 470856, 6641793, 105398650, 1858413061, 36044759796, 762659322385, 17481598316742, 431535346662645, 11413394655983536, 321989729198400385, 9651573930139850610, 306321759739045148293, 10262156907184058219340

Offset: 0

Vaclav Kotesovec, Jun 16 2018

nonn

Cf. A305133, A006153, A009306, A009444, A072597, A089148, A302397.

nmax = 20; CoefficientList[Series[(1+x)/(E^(-x)-x), {x, 0, nmax}], x] * Range[0, nmax]!
a={1};For[n=1,n<20,n++,AppendTo[a,Sum[(n!)*((n-k+1)^(k-1))*(n+1)/(k!),{k,0,n+1}]]]; a (* Detlef Meya, Sep 05 2023 *)

a(n) ~ n! / LambertW(1)^(n+1).
a(n) = (-1)^n * A009444(n+1).
a(n) = Sum_{k=0..n+1} (n+1)!*(n-k+1)^(k-1)/k! for n > 0. - Detlef Meya, Sep 05 2023

A368265 Expansion of e.g.f. exp(2x) / (1 - xexp(x)).

Original entry on oeis.org

1, 3, 12, 65, 460, 4057, 42922, 529769, 7472808, 118586033, 2090936014, 40554647377, 858082563532, 19668880007129, 485528656965762, 12841428220413593, 362276791422785488, 10859170086870710497, 344648459867067117334, 11546148650974694099201

Offset: 0

Seiichi Manyama, Dec 19 2023

Cf. A006153, A072597, A092148, A368266.

Cf. A368267, A368271.

PARI
```
a(n) = n!*sum(k=0, n, (n-k+2)^k/k!);
```

a(n) = n! * Sum_{k=0..n} (n-k+2)^k / k!.

a(n) ~ n! / ((1 + LambertW(1)) * LambertW(1)^(n+2)). - Vaclav Kotesovec, Dec 29 2023

A379933 Expansion of e.g.f. 1/( exp(-x) - x )^2.

Original entry on oeis.org

1, 4, 22, 158, 1408, 15002, 186100, 2634998, 41937136, 741170834, 14402727484, 305225470046, 7005711916840, 173134991854970, 4583675648417044, 129424786945875398, 3882446011526729440, 123304773913531035170, 4133369745467043807340, 145840627118145774415214

Offset: 0

Seiichi Manyama, Jan 06 2025

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..410

Cf. A072597, A379942, A379943.
Cf. A379934, A379936.
Cf. A358738, A377529.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x)^2))

a(n) = n!*sum(k=0, n, (k+1)*(k+2)^(n-k)/(n-k)!);

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A072597.

a(n) = n! * Sum_{k=0..n} (k+1) * (k+2)^(n-k)/(n-k)!.

A092148 Expansion of e.g.f. 1/(exp(x)-xexp(2x)).

Original entry on oeis.org

1, 0, 3, 11, 85, 739, 7831, 96641, 1363209, 21632759, 381433771, 7398080029, 156533563693, 3588046200179, 88571349871551, 2342565398442569, 66087436823953681, 1980956920420309231, 62871632567144951635, 2106277265332074827573, 74276723394195659799861

Offset: 0

Ralf Stephan, Mar 31 2004

nonn

Cf. A006153, A072597.

With[{nn=20},CoefficientList[Series[1/(Exp[x]-x Exp[2x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 19 2020 *)

PARI
```
a(n)=n!*sum(k=0,n,(n-k-1)^k/k!)
```

a(n) = n! * Sum_{k=0..n} (n-k-1)^k/k!. [Corrected by Georg Fischer, Jun 22 2022]

a(n) ~ n! / ((LambertW(1) + 1) * LambertW(1)^(n-1)). - Vaclav Kotesovec, Jun 22 2022

Corrected and extended by Harvey P. Dale, Sep 19 2020

A336948 E.g.f.: 1 / (exp(-3*x) - x).

Original entry on oeis.org

1, 4, 23, 195, 2229, 31863, 546255, 10925757, 249753897, 6422808411, 183524701779, 5768419379913, 197791542799965, 7347180526444359, 293912722687075767, 12597352573293062757, 575928946256877156177, 27976119070974574461363, 1438896686251112024068251

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Cf. A072597, A328182, A336947, A336949, A336951.

nmax = 18; CoefficientList[Series[1/(Exp[-3 x] - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(3 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 18}]
a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] - Sum[Binomial[n, k] (-3)^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 18}]

seq(n)={ Vec(serlaplace(1 / (exp(-3*x + O(x*x^n)) - x))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (3 * (n-k+1))^k / k!.
a(0) = 1; a(n) = 4 * n * a(n-1) - Sum_{k=2..n} binomial(n,k) * (-3)^k * a(n-k).
a(n) ~ n! / ((1 + LambertW(3)) * (LambertW(3)/3)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

A379943 Expansion of e.g.f. 1/( exp(-x) - x )^4.

Original entry on oeis.org

1, 8, 76, 844, 10776, 155844, 2520856, 45125924, 886037216, 18938440324, 437820992136, 10886467502244, 289738784758096, 8218731027307844, 247539834718198136, 7889896358130120484, 265325716114102815936, 9388476560982511842564, 348703400008471862936296

Offset: 0

Seiichi Manyama, Jan 07 2025

nonn

Cf. A072597, A379933, A379942.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x)^4))

a(n) = n!*sum(k=0, n, (k+4)^(n-k)*binomial(k+3, 3)/(n-k)!);

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A072597.

a(n) = n! * Sum_{k=0..n} (k+4)^(n-k) * binomial(k+3,3)/(n-k)!.

A266329 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(x) * exp( Integral A(x) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, 1, 3, 12, 62, 395, 2994, 26331, 263729, 2964845, 36975858, 506687604, 7568226163, 122388728056, 2130425343621, 39718373337525, 789613850257051, 16674806980716514, 372771700023167862, 8794945626017009781, 218392778569695964100, 5693513850197410142081, 155482323312112362743373, 4438621019461797437443233, 132210153223378852014571364, 4101859859297789141335079684, 132343983668857026899533814277

Offset: 0

Paul D. Hanna, Jan 24 2016

nonn

Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x).
What is Limit (a(n)/n!)^(1/n) ?  Example: (a(300)/300!)^(1/300) = 1.2409703...
Limit (a(n)/n!)^(1/n) = 1/Integral_{x=0..infinity} 1/(x + exp(x)) dx = 1.24008610649849766623949... - Vaclav Kotesovec, Aug 21 2017

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 62*x^4/4! + 395*x^5/5! + 2994*x^6/6! + 26331*x^7/7! + 263729*x^8/8! + 2964845*x^9/9! + 36975858*x^10/10! +...
such that log(A(x)) = Integral B(x) dx
where
B(x) = 1 + 2*x + 5*x^2/2! + 17*x^3/3! + 79*x^4/4! + 474*x^5/5! + 3468*x^6/6! + 29799*x^7/7! + 293528*x^8/8! + 3258373*x^9/9! + 40234231*x^10/10! +...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) - 1,
(2) B(x) = A'(x)/A(x),
(3) B(x) = A(x) + log(A(x)),
(4) log(A(x)) = Integral B(x) dx,
(5) log(B(x)) = Integral A(x) dx + x.
The Series Reversion of log(A(x)) equals Integral 1/(exp(x) + x) dx:
Integral 1/(exp(x) + x) dx  =  x - 2*x^2/2! + 7*x^3/3! - 37*x^4/4! + 261*x^5/5! - 2301*x^6/6! + 24343*x^7/7! - 300455*x^8/8! + 4238153*x^9/9! - 67255273*x^10/10! +...+ (-1)^(n-1)*A072597(n-1)*x^n/n! +...
so that A( Integral 1/(exp(x) + x) dx ) = exp(x).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A266328, A072597, A289739.

a[ n_] := a[n] = If[ n < 1, Boole[n == 0], Sum[ Binomial[n - 1, k - 1] a[n - k] Sum[ a[k - j], {j, k}], {k, n}]]; (* Michael Somos, Aug 08 2017 *)

{a(n) = my(A=1+x,B=1+x); for(i=0,n, A = exp( intformal( B + x*O(x^n) ) ); B = exp( intformal( 1 + A ) ) ); n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

{a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) + x) ) )), n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f. A(x) satisfies:
(1) A(x) = exp( Integral A(x) + log(A(x)) dx ).
(2) A(x) = A'(x)/A(x) - log(A(x)).
(3) log(A(x)) = exp(x) * Integral exp(-x)*A(x) dx.
(4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) + x) dx ) ).
a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..infinity} 1/(x + exp(x)) dx = 1.2400861064984976662394901721056528110217273471501174317019052800276... - Vaclav Kotesovec, Aug 21 2017

A336947 E.g.f.: 1 / (exp(-2*x) - x).

Original entry on oeis.org

1, 3, 14, 98, 920, 10792, 151888, 2494032, 46803072, 988095104, 23178247424, 598074306304, 16835199087616, 513385352524800, 16859837094942720, 593234633904293888, 22265289445252628480, 887889931920920313856, 37489832605652634763264, 1670894259596134872711168

Offset: 0

Ilya Gutkovskiy, Aug 08 2020

nonn

Cf. A072597, A216794, A336948, A336949, A336950.

nmax = 19; CoefficientList[Series[1/(Exp[-2 x] - x), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(2 (n - k + 1))^k/k!, {k, 0, n}], {n, 0, 19}]
a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] - Sum[Binomial[n, k] (-2)^k a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 19}]

seq(n)={ Vec(serlaplace(1 / (exp(-2*x + O(x*x^n)) - x))) } \\ Andrew Howroyd, Aug 08 2020

a(n) = n! * Sum_{k=0..n} (2 * (n-k+1))^k / k!.
a(0) = 1; a(n) = 3 * n * a(n-1) - Sum_{k=2..n} binomial(n,k) * (-2)^k * a(n-k).
a(n) ~ n! / ((1 + LambertW(2)) * (LambertW(2)/2)^(n+1)). - Vaclav Kotesovec, Aug 09 2021

cp's OEIS Frontend

A006153 E.g.f.: 1/(1-x*exp(x)).

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A152818 Array read by antidiagonals: A(n,k) = (k+1)^n*(n+k)!/n!.

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Magma

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A305990 Expansion of e.g.f.: (1+x) / (exp(-x) - x).

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Mathematica

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A368265 Expansion of e.g.f. exp(2*x) / (1 - x*exp(x)).

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PARI

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A379933 Expansion of e.g.f. 1/( exp(-x) - x )^2.

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PARI

PARI

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A092148 Expansion of e.g.f. 1/(exp(x)-x*exp(2*x)).

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Mathematica

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Extensions

A336948 E.g.f.: 1 / (exp(-3*x) - x).

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Mathematica

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Formula

A379943 Expansion of e.g.f. 1/( exp(-x) - x )^4.

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A368265 Expansion of e.g.f. exp(2x) / (1 - xexp(x)).

A092148 Expansion of e.g.f. 1/(exp(x)-xexp(2x)).