A001861 -id:A001861 - cp's OEIS frontend

A343523 a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(n,k) * a(k-1).

1, 2, 8, 34, 164, 878, 5136, 32490, 220476, 1594470, 12223016, 98876322, 840804820, 7491247006, 69730182720, 676390547034, 6821988655468, 71398971351510, 774032400213336, 8677733804696594, 100459693769214980, 1199306075189097230, 14746332963835756400, 186534818943430728906

Offset: 0

Ilya Gutkovskiy, Jun 07 2021

nonn

Georg Fischer, Table of n, a(n) for n = 0..400

Cf. A001861, A004123, A035009, A040027, A343975, A344735, A344840, A345077, A345078, A345081.

a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n, k] a[k - 1], {k, 1, n}]; Table[a[n], {n, 0, 23}]
nmax = 23; A[] = 0; Do[A[x] = 1 + 2 x A[x/(1 - x)]/(1 - x)^2 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + 2 * x * A(x/(1 - x)) / (1 - x)^2.

A355291 Expansion of e.g.f. exp(exp(x)*(exp(x) + 1) - 2).

Original entry on oeis.org

1, 3, 14, 81, 551, 4266, 36803, 348543, 3583484, 39652659, 468970211, 5894584812, 78366374813, 1097537989671, 16136598952718, 248309032411485, 3988468487017379, 66715970326561170, 1159712730763363991, 20909709414253764819, 390374806223071148084, 7534929383736826736007

Offset: 0

Vaclav Kotesovec, Jun 27 2022

nonn

In general, if m > 0, b > d >= 1 and e.g.f. = exp(m*exp(b*x) + r*exp(d*x) + s) then a(n) ~ exp(m*exp(b*z) + r*exp(d*z) + s - n) * (n/z)^(n + 1/2) / sqrt(m*b*(1 + b*z)*exp(b*z) + r*d*(1 + d*z)*exp(d*z)), where z = LambertW(n/m)/b - 1/(d + b/LambertW(n/m) + b^2 * m^(d/b) * n^(1 - d/b) * (1 + LambertW(n/m)) / (d*r*LambertW(n/m)^(2 - d/b))). - Vaclav Kotesovec, Jul 03 2022

In addition, if b/d >=2 then a(n) ~ c * (b*n/LambertW(n/m))^n * exp(n/LambertW(n/m) + r * (n/(m*LambertW(n/m)))^(d/b) - n + s) / sqrt(1 + LambertW(n/m)), where c = 1 for b/d > 2 and c = exp(-r^2/(8*m)) for b/d = 2. - Vaclav Kotesovec, Jul 10 2022

Seiichi Manyama, Table of n, a(n) for n = 0..500
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.

Cf. A001861, A055882, A126390, A143405, A355379.

nmax = 20; CoefficientList[Series[Exp[Exp[2*x] - 2 + Exp[x]], {x, 0, nmax}], x] * Range[0, nmax]!
Table[Sum[Binomial[n, k] * 2^k * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^30)); Vec(serlaplace(exp(exp(x)*(exp(x) + 1) - 2))) \\ Michel Marcus, Jun 27 2022

a(n) = Sum_{k=0..n} binomial(n,k) * 2^k * Bell(k) * Bell(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} (1 + 2^k) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Jul 01 2022
a(n) ~ exp(exp(2*z) + exp(z) - 2 - n) * (n/z)^(n + 1/2) / sqrt(2*(1 + 2*z)*exp(2*z) + (1 + z)*exp(z)), where z = LambertW(n)/2 - 1/(1 + 2/LambertW(n) + 4 * n^(1/2) * (1 + LambertW(n)) / LambertW(n)^(3/2)). - Vaclav Kotesovec, Jul 03 2022
a(n) ~ 2^n * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n * exp(n + 17/8 - n/LambertW(n) - sqrt(n/LambertW(n)))). - Vaclav Kotesovec, Jul 08 2022

A001862 Number of forests of least height with n nodes.

Original entry on oeis.org

1, 1, 2, 7, 26, 111, 562, 3151, 19252, 128449, 925226, 7125009, 58399156, 507222535, 4647051970, 44747776651, 451520086208, 4761032807937, 52332895618066, 598351410294193, 7102331902602676, 87365859333294151, 1111941946738682522, 14621347433458883187

Offset: 0

N. J. A. Sloane

nonn

From Gus Wiseman, Feb 14 2024: (Start)
Also the number of minimal loop-graphs covering n vertices. This is the minimal case of A322661. For example, the a(0) = 1 through a(3) = 7 loop-graphs are (loops represented as singletons):
  {}  {1}  {12}   {1-23}
           {1-2}  {2-13}
                  {3-12}
                  {12-13}
                  {12-23}
                  {13-23}
                  {1-2-3}
(End)

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. See (3.3.7): number of ways to cover the complete graph K_n with star graphs.
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
John Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.
John Riordan, Letter to N. J. A. Sloane, Oct. 1970
John Riordan and N. J. A. Sloane, Correspondence, 1974

The connected case is A000272.
Without loops we have A053530, minimal case of A369191.
This is the minimal case of A322661.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts simple graphs; also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
Cf. A000085, A000169, A003465, A062740, A066383, A133686, A368597.

Range[0, 20]! CoefficientList[Series[Exp[x Exp[x] - x^2/2], {x, 0, 20}], x] (* Geoffrey Critzer, Mar 13 2011 *)
fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]& /@ Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length@fasmin[Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&]],{n,0,4}] (* Gus Wiseman, Feb 14 2024 *)

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

Binomial transform of A053530. - Gus Wiseman, Feb 14 2024

Formula and more terms from Vladeta Jovovic, Mar 27 2001

A036075 The number of partitions of {1..5n} that are invariant under a permutation consisting of n 5-cycles.

Original entry on oeis.org

1, 2, 10, 70, 602, 6078, 70402, 917830, 13253002, 209350350, 3584098770, 66012131222, 1300004931162, 27232369503902, 604103160535330, 14136908333006822, 347827448896896554, 8971450949011952494

Offset: 0

N. J. A. Sloane

nonn

Original name: Sorting numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A001861, A002872-A002875, A036074.
u[n,j] generates for j=1, A000110 Bell numbers; j=2, A002872; j=3, A002874; j=4, A141003 (Mathar); j=5, this sequence; j=6, A141004 (Mathar); j=7, A036077. - Wouter Meeussen, Dec 06 2008
Column 5 of A162663.

u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; Table[u[n,5],{n,0,12}] (* Wouter Meeussen, Dec 06 2008 *)
mx = 16; p = 5; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 5^k * BellB[k, 1/5] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=5.
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=5. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (5*n/LambertW(5*n))^n * exp(n/LambertW(5*n) + (5*n/LambertW(5*n))^(1/5) - n - 6/5) / sqrt(1 + LambertW(5*n)). - Vaclav Kotesovec, Jul 10 2022

New name from Danny Rorabaugh, Oct 24 2015

A036077 The number of partitions of {1..7n} that are invariant under a permutation consisting of n 7-cycles.

Original entry on oeis.org

1, 2, 12, 106, 1144, 14434, 209736, 3451290, 63194936, 1269555762, 27700698344, 651497885482, 16414347638936, 440651469115394, 12546081858835528, 377328994871025210, 11946046637611280120

Offset: 0

N. J. A. Sloane

nonn

Original name: Sorting numbers.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
Index entries for sequences related to sorting

Cf. A001861, A002872-A002875, A036074.
u[n,j] generates for j=1, A000110; j=2, A002872; j=3, A002874; j=4, A141003; j=5, A036075; j=6, A141004; j=7, this sequence. - Wouter Meeussen, Dec 06 2008
Column 7 of A162663.

u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; Table[u[n,7],{n,0,12}] (* Wouter Meeussen, Dec 06 2008 *)
mx = 16; p = 7; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* Robert G. Wilson v, Dec 12 2012 *)
Table[Sum[Binomial[n,k] * 7^k * BellB[k, 1/7] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 29 2022 *)

E.g.f.: exp((exp(p*x)-p-1)/p+exp(x)) for p=7.
a(n) ~ exp(exp(p*r)/p + exp(r) - 1 - 1/p - n) * (n/r)^(n + 1/2) / sqrt((1 + p*r)*exp(p*r) + (1 + r)*exp(r)), where r = LambertW(p*n)/p - 1/(1 + p/LambertW(p*n) + n^(1 - 1/p) * (1 + LambertW(p*n)) * (p/LambertW(p*n))^(2 - 1/p)) for p=7. - Vaclav Kotesovec, Jul 03 2022
a(n) ~ (7*n/LambertW(7*n))^n * exp(n/LambertW(7*n) + (7*n/LambertW(7*n))^(1/7) - n - 8/7) / sqrt(1 + LambertW(7*n)). - Vaclav Kotesovec, Jul 10 2022

New name from Danny Rorabaugh, Oct 24 2015

A339014 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2)).

Original entry on oeis.org

1, 0, 0, 2, 2, 2, 42, 142, 366, 3082, 18626, 86990, 596158, 4485626, 30214498, 224897662, 1871664190, 15587540042, 134045407458, 1231183979886, 11725017784574, 114812031304986, 1170100796863202, 12371771640238174, 134796972965052350, 1514854948728869354

Offset: 0

Ilya Gutkovskiy, Nov 19 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..563

Cf. A001861, A006505, A194689.

nmax = 25; CoefficientList[Series[Exp[2 (Exp[x] - 1 - x - x^2/2)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 25}]

my(x='x+O('x^30)); Vec(serlaplace(exp(2 * (exp(x) - 1 - x - x^2/2)))) \\ Michel Marcus, Nov 19 2020

a(0) = 1; a(n) = 2 * Sum_{k=3..n} binomial(n-1,k-1) * a(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A006505(k) * A006505(n-k).

A033306 Triangle of coefficients of ordered cycle-index polynomials: T(n,k) = binomial(n,k)Bell(k)Bell(n-k).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 6, 6, 5, 15, 20, 24, 20, 15, 52, 75, 100, 100, 75, 52, 203, 312, 450, 500, 450, 312, 203, 877, 1421, 2184, 2625, 2625, 2184, 1421, 877, 4140, 7016, 11368, 14560, 15750, 14560, 11368, 7016, 4140, 21147, 37260, 63144, 85260, 98280, 98280, 85260, 63144, 37260, 21147

Offset: 0

N. J. A. Sloane

			   1;
   1,  1;
   2,  2,   2;
   5,  6,   6,   5;
  15, 20,  24,  20, 15;
  52, 75, 100, 100, 75, 52;
  ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 80.

Cf. A000110, row sums give A001861.
Columns include A000110 and A052889.
Cf. A000807.

A033306 := proc(n,k)
    if k < 0 or k > n then
        0;
    else
        binomial(n,k)*combinat[bell](k)*combinat[bell](n-k) ;
    end if;
end proc: # R. J. Mathar, Mar 21 2013
# second Maple program:
b:= proc(n) option remember; expand(`if`(n>0, add(
     (x^j+1)*b(n-j)*binomial(n-1, j-1), j=1..n), 1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 30 2019

t[n_, k_] := Binomial[n, k] * BellB[k] * BellB[n-k]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 14 2014 *)

E.g.f.: exp(exp(x*y)+exp(x)-2).

Sum_{k=0..2n} (-1)^k * T(2n,k) = A000807(n). - Alois P. Heinz, Feb 13 2024

Edited by Vladeta Jovovic, Sep 17 2003

A059099 Expansion of exp(exp(x)-1)/(2-exp(x)).

Original entry on oeis.org

1, 2, 7, 33, 198, 1453, 12669, 128320, 1482721, 19260421, 277913552, 4410640919, 76360030701, 1432144732762, 28926138244883, 625974400305541, 14449445989893990, 354384475357492593, 9202837263156670345, 252260867710562944224, 7278710072406887897461

Offset: 0

Vladeta Jovovic, Jan 02 2001

Row sums of A227343. - Peter Bala, Jul 11 2013

The sequence gives the number of barred preferential arrangements of an n-set having one bar, where one fixed section is a free section and elements which are to go into the other section are partitioned into unordered nonempty subsets. - Sithembele Nkonkobe, Jul 02 2015

			exp(exp(x)-1)/(2-exp(x)) = 1 + 2*x + 7/2*x^2 + 11/2*x^3 + 33/4*x^4 + 1453/120*x^5 + 4223/240*x^6 + 1604/63*x^7 + ...

Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 12, 19, 29.
S. Nkonkobe and V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO], 2015.

Row sums of A059098.

Cf. A001861, A000110, A005493, A049020, A227343.

s := series(exp(exp(x)-1)/(2-exp(x)), x, 60): for i from 0 to 50 do printf(`%d,`,i!*coeff(s,x,i)) od:

CoefficientList[Series[E^(E^x-1)/(2-E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 02 2015 *)

a(n) = Sum_{m=0..n} Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..m} (i-j+1).
Stirling transform of A000522. - Vladeta Jovovic, May 10 2004
a(n) ~ n!*exp(1)/(2*(log(2))^(n+1)). - Vaclav Kotesovec, Jul 02 2015

More terms from James Sellers, Jan 03 2001

A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).

Original entry on oeis.org

1, 2, 9, 58, 485, 4986, 60877, 861554, 13878153, 250854130, 5030058161, 110837000682, 2662669300909, 69270266115818, 1940260799150325, 58220372514830626, 1863293173842259217, 63356877145370671074

Offset: 0

Vladeta Jovovic, Jan 06 2001

L'(n,i) are unsigned Lah numbers (Cf. A008297): L'(n,i) = (n!/i!)*binomial(n-1,i-1) for i >= 1, L'(0,0) = 1, L'(n,0) = 0 for n > 0.

			(1-x)/(1-2*x)*exp(x/(1-x)) = 1 + 2*x + 9/2*x^2 + 29/3*x^3 + 485/24*x^4 + 831/20*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 0..250
Index entries for sequences related to Laguerre polynomials

Cf. A001861, A049020, A052897, A059099, A059110.

Row sums of A059114.

[Factorial(n)*(&+[Evaluate(LaguerrePolynomial(n-k, k-1), -1) : k in [0..n]]): n in [0..30]]; // G. C. Greubel, Feb 23 2021

s := series((1-x)/(1-2*x)*exp(x/(1-x)), x, 21): for i from 0 to 20 do printf(`%d,`,i!*coeff(s,x,i)) od:

With[{nn=20},CoefficientList[Series[(1-x)/(1-2x) Exp[x/(1-x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 18 2020 *)
Table[n!*Sum[LaguerreL[n-k, k-1, -1], {k,0,n}], {n,0,30}] (* G. C. Greubel, Feb 23 2021 *)

{a(n)=if(n<0, 0, n!*polcoeff( (1-x)/(1-2*x)*exp(x/(1-x)+x*O(x^n)), n))} /* Michael Somos, Aug 03 2006 */

{a(n)=local(A); if(n<0,0, n++; A=vector(n); A[n]=1; for(k=1,n-1, A[n-k]=1; if(k>1, A[n-k+1]=A[n-k+2]); for(i=n-k+1,n, A[i]=A[i-1]+k*A[i])); A[n])} /* Michael Somos, Aug 03 2006 */

a(n) = n!*sum(k=0, n, pollaguerre(n-k, k-1, -1)); \\ Michel Marcus, Feb 23 2021

[factorial(n)*sum( gen_laguerre(n-k, k-1, -1) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Feb 23 2021

Sum_{m=0..n} Sum_{i=0..n} L'(n, i)*Product_{j=1..m} (i-j+1).
Given g.f. A(x), then g.f. A000522 = A(x/(1+x)). - Michael Somos, Aug 03 2006
a(n) = n!*Sum_{k=0..n} LaguerreL(n-k, k-1, -1). - G. C. Greubel, Feb 23 2021
a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n + 1/2) / exp(n-1). - Vaclav Kotesovec, Feb 23 2021

Definition clarified by Harvey P. Dale, Jul 18 2020

A346417 E.g.f.: exp(exp(2*(exp(x) - 1)) - 1).

Original entry on oeis.org

1, 2, 10, 66, 538, 5186, 57402, 714594, 9853978, 148774914, 2436823034, 42979319202, 811254807770, 16302732719682, 347248840767162, 7809649226242530, 184831773033020826, 4589793199157616770, 119272846472231229818, 3235960069037751550498, 91466308730323104617050

Offset: 0

Ilya Gutkovskiy, Jul 16 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..449

Cf. A000258, A000898, A001861, A055882, A126390, A136658.

b:= proc(n, t, m) option remember; `if`(n=0, `if`(t=1, 1,
      b(m, 1, 0)*2^m) , m*b(n-1, t, m)+b(n-1, t, m+1))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 06 2021

nmax = 20; CoefficientList[Series[Exp[Exp[2 (Exp[x] - 1)] - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 2^k BellB[k], {k, 0, n}], {n, 0, 20}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] BellB[k, 2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(2*(exp(x) - 1)) - 1))) \\ Michel Marcus, Jul 19 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * 2^k * Bell(k).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001861(k) * a(n-k).

cp's OEIS Frontend

A343523 a(0) = 1; a(n) = 2 * Sum_{k=1..n} binomial(n,k) * a(k-1).

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A355291 Expansion of e.g.f. exp(exp(x)*(exp(x) + 1) - 2).

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A001862 Number of forests of least height with n nodes.

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A036075 The number of partitions of {1..5n} that are invariant under a permutation consisting of n 5-cycles.

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Mathematica

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A036077 The number of partitions of {1..7n} that are invariant under a permutation consisting of n 7-cycles.

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Mathematica

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A339014 E.g.f.: exp(2 * (exp(x) - 1 - x - x^2 / 2)).

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A033306 Triangle of coefficients of ordered cycle-index polynomials: T(n,k) = binomial(n,k)*Bell(k)*Bell(n-k).

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A059099 Expansion of exp(exp(x)-1)/(2-exp(x)).

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A033306 Triangle of coefficients of ordered cycle-index polynomials: T(n,k) = binomial(n,k)Bell(k)Bell(n-k).

A059115 Expansion of e.g.f.: ((1-x)/(1-2x))exp(x/(1-x)).