A006152 -id:A006152 - 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

A006154 Number of labeled ordered partitions of an n-set into odd parts.

Original entry on oeis.org

1, 1, 2, 7, 32, 181, 1232, 9787, 88832, 907081, 10291712, 128445967, 1748805632, 25794366781, 409725396992, 6973071372547, 126585529106432, 2441591202059281, 49863806091395072, 1074927056650469527, 24392086908129247232, 581176736647853024581

Offset: 0

Simon Plouffe

Conjecture: taking the sequence modulo an integer k gives an eventually periodic sequence. For example, the sequence taken modulo 10 is [1, 1, 2, 7, 2, 1, 2, 7, 2, 1, 2, 7, 2, ...], with an apparent period [1, 2, 7, 2] beginning at a(1), of length 4. Cf. A000670. - Peter Bala, Apr 12 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Farid Aliniaeifard and Shu Xiao Li, Peak algebra in noncommuting variables, arXiv:2506.12868 [math.CO], 2025. See pp. 2, 9, 45.
Philippe Flajolet, Symbolic Enumerative Combinatorics and Complex Asymptotic Analysis, Algorithms Seminar 2000-2001, F. Chyzak (ed.), INRIA, (2002), pp. 161-170.
S. Getu and L. W. Shapiro, Combinatorial view of the composition of functions, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
Prabha Sivaraman Nair and Rejikumar Karunakaran, On k-Fibonacci Brousseau Sums, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 8.

Cf. A000045, A000111, A000670, A003724, A193474, A196776.

readlib(coeftayl):
with(combinat, bell);
A:=series(1/(1-sinh(x)),x,20);
G(x):=A : f[0]:=G(x): for n from 0 to 21 do f[n]:=coeftayl(G(x), x=0, n);;
p[n]:=f[n]*((n)!) od: x:=0:seq(p[n], n=0..20); # Sergei N. Gladkovskii, Jun 01 2012
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add((i->
      a(n-i)*binomial(n, i))(2*j+1), j=0..(n-1)/2))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 01 2022

a[n_] := Sum[ (-1)^i*(k - 2*i)^n*Binomial[k, i]/2^k, {k, 1, n}, {i, 0, k}]; a[0] = 1; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Dec 07 2011, after Vladimir Kruchinin *)
With[{nn=20},CoefficientList[Series[1/(1-Sinh[x]),{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Nov 16 2012 *)

a(n):=sum(sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k)/2^k,k,1,n); /* Vladimir Kruchinin, May 28 2011 */

a(n)=if(n<2,n>=0,sum(k=1,ceil(n/2),binomial(n,2*k-1)*a(n-2*k+1))) \\ Ralf Stephan

E.g.f.: 1/(1 - sinh(x)).
With alternating signs, e.g.f.: 1/(1+sinh(x)). - Ralf Stephan, Apr 29 2004
a(0) = a(1) = 1, a(n) = Sum_{k=1..ceiling(n/2)} C(n,2*k-1)*a(n-2*k+1). - Ralf Stephan, Apr 29 2004
a(n) ~ (sqrt(2)/2)*n!/log(1+sqrt(2))^(n+1). - Conjectured by Simon Plouffe, Feb 17 2007.
From Andrew Hone, Feb 22 2007: (Start)
This formula can be proved using the techniques in the article by Philippe Flajolet (see links) [see Theorem 5 and Table 2, noting that 1/(1-sinh(x)) just has a simple pole at x=log(1+sqrt(2))]. (End)
a(n) = Sum_{k=1..n} Sum_{i=0..k} (-1)^i*(k-2*i)^n*binomial(k,i)/2^k, n > 0, a(0)=1. - Vladimir Kruchinin, May 28 2011
Row sums (apart from a(0)) of A196776. - Peter Bala, Oct 06 2011
Row sums of A193474. - Peter Luschny, Oct 07 2011
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A003724 and A000111. - Peter Bala, Dec 06 2011
From Sergei N. Gladkovskii, Jun 01 2012: (Start)
Let E(x) be the e.g.f., then
E(x) = -1/x + 1/(x*(1-x))+ x^3/((1-x)*((1-x)*G(0) - x^2)); G(k) = (2*k+2)*(2*k+3)+x^2-(2*k+2)*(2*k+3)*x^2/G(k+1); (continued fraction).
E(x) = -1/x + 1/(x*(1-x))+ x^3/((1-x)*((1-x)*G(0) - x^2)); G(k) = 8*k+6+x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)); (continued fraction).
E(x) = 1/(1 - x*G(0)); G(k) = 1 + x^2/(2*(2*k+1)*(4*k+3) + 2*x^2*(2*k+1)*(4*k+3)/(-x^2 - 4*(k+1)*(4*k+5)/G(k+1))); (continued fraction).
(End).
E.g.f. 1/(1 - x*G(0)) where G(k) = 1 - x^2/( (2*k+1)*(2*k+3) - (2*k+1)*(2*k+3)^2/(2*k+3 - (2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 01 2012
O.g.f A(x) satisfies A(x) = 1 + ( A(x/(1-x))/(1-x) - A(x/(1+x))/(1+x) )/2. - Paul D. Hanna, Aug 19 2024

More terms from Christian G. Bower, Oct 15 1999

A006155 Expansion of e.g.f.: 1/(2-x-exp(x)).

Original entry on oeis.org

1, 2, 9, 61, 551, 6221, 84285, 1332255, 24066691, 489100297, 11044268633, 274327080611, 7433424980943, 218208342366093, 6898241919264181, 233651576126946103, 8441657595745501019, 324052733365292875025, 13171257161208184782225, 565092918793429218839307

Offset: 0

Simon Plouffe

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..396
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 157

Cf. A032112.

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( 1/(2-x-Exp(x)) ))); // G. C. Greubel, Jan 09 2025

With[{nn=20},CoefficientList[Series[1/(2-x-E^x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 27 2018 *)

def A006155_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1/(2-x-exp(x)) ).egf_to_ogf().list()
print(A006155_list(40)) # G. C. Greubel, Jan 09 2025

E.g.f.: 1/(2-x-exp(x)).
a(n) ~ n! / ((1+c) * (2-c)^(n+1)), where c = A226571 = LambertW(exp(2)). - Vaclav Kotesovec, Jun 06 2019
a(0) = 1; a(n) = n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020

More terms from Ralf Stephan, Mar 12 2004

A206703 Triangular array read by rows. T(n,k) is the number of partial permutations (injective partial functions) of {1,2,...,n} that have exactly k elements in a cycle. The k elements are not necessarily in the same cycle. A fixed point is considered to be in a cycle.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 13, 9, 6, 6, 73, 52, 36, 24, 24, 501, 365, 260, 180, 120, 120, 4051, 3006, 2190, 1560, 1080, 720, 720, 37633, 28357, 21042, 15330, 10920, 7560, 5040, 5040, 394353, 301064, 226856, 168336, 122640, 87360, 60480, 40320, 40320

Offset: 0

Geoffrey Critzer, Feb 11 2012

			     1;
     1,     1;
     3,     2,     2;
    13,     9,     6,     6;
    73,    52,    36,    24,    24;
   501,   365,   260,   180,   120,  120;
  4051,  3006,  2190,  1560,  1080,  720,   720;
  ...

Mohammad K. Azarian, On the Fixed Points of a Function and the Fixed Points of its Composite Functions, International Journal of Pure and Applied Mathematics, Vol. 46, No. 1, 2008, pp. 37-44. Mathematical Reviews, MR2433713 (2009c:65129), March 2009. Zentralblatt MATH, Zbl 1160.65015.
Mohammad K. Azarian, Fixed Points of a Quadratic Polynomial, Problem 841, College Mathematics Journal, Vol. 38, No. 1, January 2007, p. 60. Solution published in Vol. 39, No. 1, January 2008, pp. 66-67.

Alois P. Heinz, Rows n = 0..140, flattened
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 132.

Columns k = 0..1 give: A000262, A006152.
Main diagonal gives A000142.
Row sums give A002720.
T(2n,n) gives A088026.
Cf. A002720, A052852, A103194, A105219, A331725.

b:= proc(n) option remember; `if`(n=0, 1, add((p-> p+x^j*
      coeff(p, x, 0))(b(n-j)*binomial(n-1, j-1)*j!), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 19 2022

nn = 7; a = 1/(1 - x); ay = 1/(1 - y x); f[list_] := Select[list, # > 0 &]; Map[f, Range[0, nn]! CoefficientList[Series[Exp[a x] ay, {x, 0, nn}], {x, y}]] // Flatten

E.g.f.: exp(x/(1-x))/(1-y*x).
From Alois P. Heinz, Feb 19 2022: (Start)
Sum_{k=1..n} T(n,k) = A052852.
Sum_{k=0..n} k * T(n,k) = A103194(n).
Sum_{k=0..n} (n-k) * T(n,k) =  A105219(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A331725(n). (End)

A320264 Number T(n,k) of proper multisets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the multiset; triangle T(n,k), n>=2, 1<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 3, 11, 9, 4, 38, 84, 52, 7, 125, 523, 766, 365, 10, 364, 2676, 7096, 7775, 3006, 16, 1041, 12435, 52955, 100455, 87261, 28357, 22, 2838, 54034, 348696, 1020805, 1497038, 1074766, 301064, 32, 7645, 225417, 2120284, 8995801, 19823964, 23605043, 14423564, 3549177

Offset: 2

Alois P. Heinz, Oct 08 2018

			T(2,1) = 1: {a,a}.
T(3,2) = 2: {a,a,b}, {a,b,b}.
T(4,3) = 9: {a,a,b,c}, {a,a,bc}, {a,a,cb}, {b,b,a,c}, {b,b,ac}, {b,b,ca}, {c,c,a,b}, {c,c,ab}, {c,c,ba}.
Triangle T(n,k) begins:
  .
  .   .
  .   1,    .
  .   1,    2,     .
  .   3,   11,     9,      .
  .   4,   38,    84,     52,       .
  .   7,  125,   523,    766,     365,       .
  .  10,  364,  2676,   7096,    7775,    3006,       .
  .  16, 1041, 12435,  52955,  100455,   87261,   28357,      .
  .  22, 2838, 54034, 348696, 1020805, 1497038, 1074766, 301064,   .

Alois P. Heinz, Rows n = 2..150

Column k=1 gives A047967.
Row sums give A320265.
T(n+1,n) gives A006152.
Cf. A257740, A319501.

h:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(h(n-i*j, i-1, k)*binomial(k^i, j), j=0..n/i)))
    end:
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(
      d*k^d, d=numtheory[divisors](j))*g(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> add((-1)^i*(g(n, k-i)-h(n$2, k-i))*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=1..n-1), n=2..12);

h[n_, i_, k_] := h[n, i, k] = If[n == 0, 1, If[i<1, 0,
     Sum[h[n - i*j, i-1, k]*Binomial[k^i, j], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[Sum[
     d*k^d, {d, Divisors[j]}]*g[n - j, k], {j, 1, n}]/n];
T[n_, k_] := Sum[(-1)^i*(g[n, k-i]-h[n, n, k-i])*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, 1, n - 1}], {n, 2, 12}] // Flatten (* Jean-François Alcover, Feb 13 2021, after Alois P. Heinz *)

T(n,k) = A257740(n,k) - A319501(n,k).

A114329 Triangle T(n,k) is the number of partitions of an n-set into lists (cf. A000262) with k lists of size 1.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 36, 24, 12, 0, 1, 240, 180, 60, 20, 0, 1, 1920, 1440, 540, 120, 30, 0, 1, 17640, 13440, 5040, 1260, 210, 42, 0, 1, 183120, 141120, 53760, 13440, 2520, 336, 56, 0, 1, 2116800, 1648080, 635040, 161280, 30240, 4536, 504, 72, 0, 1

Offset: 0

Vladeta Jovovic, Feb 06 2006

The average number of size 1 lists goes to 1 as n->infinity. In other words, lim_{n->infinity} Sum_{k>=1} T(n,k)*k/A000262(n) = 1. - Geoffrey Critzer, Feb 20 2022 (after asymptotic limits by Vaclav Kotesovec given in A000262)

			Triangle begins:
    1;
    0,   1;
    2,   0,  1;
    6,   6,  0,  1;
   36,  24, 12,  0, 1;
  240, 180, 60, 20, 0, 1;
  ...

Nathaniel Johnston, Table of n, a(n) for n = 0..5150

Cf. A000262, A006152, A052845, A052852.

t:=taylor(exp(x/(1-x)+(y-1)*x),x,11):for n from 0 to 10 do for k from 0 to n do printf("%d, ",coeff(n!*coeff(t,x,n),y,k)): od: printf("\n"): od: # Nathaniel Johnston, Apr 27 2011
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*
     `if`(j=1, x, 1)*b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 19 2022

nn = 10; Table[Take[(Range[0, nn]! CoefficientList[ Series[Exp[ x/(1 - x) - x + y x], {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}] // Grid (* Geoffrey Critzer, Feb 19 2022 *)

E.g.f.: exp(x/(1-x)+(y-1)*x). More generally, e.g.f. for number of partitions of n-set into lists with k lists of size m is exp(x/(1-x)+(y-1)*x^m).

A300159 Number of ways of converting one set of lists containing n elements to another set of lists containing n elements by removing the last element from one of the lists and either appending it to an existing list or treating it as a new list.

Original entry on oeis.org

0, 0, 4, 30, 240, 2140, 21300, 235074, 2853760, 37819800, 543445380, 8416452550, 139753069104, 2476581106740, 46648575724660, 930581784937770, 19597766647728000, 434455097953799344, 10112163333554834820, 246539064280189932270, 6282671083849941925360

Offset: 0

Mitchell Keith Bloch, Feb 26 2018

nonn

All terms are even.

			a(0) = 0 since for 0 lists, 0 conversions are possible.
a(1) = 0 since for the 1 set of 1 list of length 1, there exist no possible conversions.
a(2) = 4 since for the 2 sets of 1 list of length 2, there exists only 1 conversion, and for the 1 set of 2 lists of length 1, there exist 2 conversions.
a(3) = 30 since for the 6 sets of 1 list of length 3, there exists 1 conversion, for the 6 sets of 1 list of length 2 and 1 list of length 1, there exist 3 conversions, and for the 1 set of 3 lists of length 1, there exists 6 conversions.

Alois P. Heinz, Table of n, a(n) for n = 0..443 (first 71 terms from Mitchell Keith Bloch)
Mitchell Keith Bloch, Program C++
Mitchell Keith Bloch, Program C++ with Boost
Index entries for sequences related to Laguerre polynomials

Extends A000262 to count conversions in addition to sets of lists.

Cf. A000041, A006152, A052852, A103194.

l:= func< n,b | Evaluate(LaguerrePolynomial(n,1), b) >;
[0,0,4]cat[Factorial(n)*( 2*l(n-2,-1) - l(n-3,-1) ): n in [3..30]]; // G. C. Greubel, Mar 09 2021

b:= proc(n, t, c) option remember; `if`(n=0, t^2-c, add(j!*
      binomial(n-1, j-1)*b(n-j, t+1, c+`if`(j=1, 1, 0)), j=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 05 2018
# second Maple program:
a:= proc(n) option remember; `if`(n<6, [0$2, 4, 30, 240, 2140][n+1],
     (n*(2*n^2-13*n+16)*a(n-1)-n*(n-1)*(n-3)*(n-4)*a(n-2))/((n-2)*(n-5)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 05 2018

(* First program *)
b[n_, t_, c_]:= b[n,t,c]= If[n==0, t^2 -c, Sum[j! Binomial[n-1, j-1]b[n-j,t+1,c + If[j==1, 1, 0]], {j,n}]];
a[n_]:= b[n, 0, 0];
a/@ Range[0, 25] (* Jean-François Alcover, Nov 24 2020, after Alois P. Heinz *)
(* Second program *)
Table[If[n<2, 0, n!*(2*LaguerreL[n-2,1,-1] -LaguerreL[n-3,1,-1])], {n,0,30}] (* G. C. Greubel, Mar 09 2021 *)

[0,0,4]+[factorial(n)*(2*gen_laguerre(n-2,1,-1) - gen_laguerre(n-3,0,-1)) for n in (3..30)] # G. C. Greubel, Mar 09 2021

a(n) = Sum_{i=1..p(n)} (n!/(Product_{j=1..n} k(i,j)!) * ((Sum_{j=1..n} k(i,j))^2 - k(i,1))) (where p(n) is the number of partitions A000041 and k(i,j) is the number of partitions of size j in partitioning i).
From Alois P. Heinz, Mar 05 2018: (Start)
E.g.f.: x^2*(2-x)*exp(x/(1-x))/(x-1)^2.
a(n) = (n*(2*n^2-13*n+16)*a(n-1) - n*(n-1)*(n-3)*(n-4)*a(n-2)) / ((n-2)*(n-5)) for n>5. (End)
a(n) ~ n^(n + 3/4) * exp(2*sqrt(n) - n - 1/2) / sqrt(2). - Vaclav Kotesovec, Jun 02 2018
a(n) = n!*( 2*LaguerreL(n-2,1,-1) - LaguerreL(n-3,1,-1) ) for n > 1, with a(0) = a(1) = 0. - G. C. Greubel, Mar 09 2021

More terms from Mitchell Keith Bloch, Mar 05 2018

A351823 Triangular array read by rows. T(n,k) is the number of sets of lists (as in A000262(n)) with exactly k size 2 lists, n >= 0, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 1, 2, 7, 6, 49, 12, 12, 301, 140, 60, 2281, 1470, 180, 120, 21211, 12642, 2940, 840, 220417, 127736, 41160, 3360, 1680, 2528569, 1527192, 455112, 70560, 15120, 32014801, 19837530, 5748120, 1234800, 75600, 30240, 442974511, 278142590, 83995560, 16687440, 1940400, 332640

Offset: 0

Geoffrey Critzer, Feb 20 2022

From the asymptotic estimate of A000262(n) provided by Vaclav Kotesovec we deduce that in the limit as n gets big the average number of size 2 lists is equal to 1. In other words, lim_{n->oo} Sum_{k>=1} T(n,k)*k/A000262(n) = 1. Generally for any j >= 1, the average number of size j lists equals 1 in the limit as n -> oo.

			Triangle T(n,k) begins:
      1;
      1;
      1,     2;
      7,     6;
     49,    12,   12;
    301,   140,   60;
   2281,  1470,  180, 120;
  21211, 12642, 2940, 840;
  ...

Column k=1 gives A113235.
T(n,floor(n/2)) gives A081125.
T(2n,n) gives A001813.
Cf. A000262 (row sums) A006152, A114329, A351825.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(j!*
     `if`(j=2, x, 1)*b(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..12);  # Alois P. Heinz, Feb 20 2022

nn = 7; Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Exp[ x/(1 - x) - x ^2 + y x^2], {x, 0, nn}], {x, y}]] // Grid

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

Sum_{k=0..floor(n/2)} k * T(n,k) = A351825(n). - Alois P. Heinz, Feb 24 2022

A351825 Total number of size 2 lists in all sets of lists partitioning [n] (cf. A000262).

Original entry on oeis.org

0, 0, 2, 6, 36, 260, 2190, 21042, 226856, 2709576, 35491770, 505620830, 7780224012, 128555409996, 2269569526406, 42625044254730, 848404205856720, 17836074466842512, 394872870912995826, 9181542826326252726, 223680717959853460340, 5697036951307194432660, 151396442683371572351742

Offset: 0

Geoffrey Critzer, Feb 20 2022

nonn

Cf. A000262, A006152, A129652, A351823.

nn = 22; Range[0, nn]! CoefficientList[Series[D[Exp[ x/(1 - x) - x ^2 + y x^2], y] /. y -> 1, {x, 0, nn}], x]
Join[{0, 0, 2}, Table[n!*Hypergeometric1F1[n-1, 2, 1]/E, {n, 3, 25}]] (* Vaclav Kotesovec, Feb 21 2022 *)

a(n) = 2*binomial(n,2)*A000262(n-2).
E.g.f.: x^2*exp(x/(1-x)) = d/dy G(x,y)|y=1 where G(x,y) is the e.g.f. for A351823.
a(n) = Sum_{k=0..floor(n/2)} k * A351823(n,k).
a(n) ~ n^(n - 1/4) * exp(2*sqrt(n) - n - 1/2) / sqrt(2) * (1 - 101/(48*sqrt(n))). - Vaclav Kotesovec, Feb 21 2022
a(n) = 2 * A129652(n,2). - Alois P. Heinz, Feb 22 2022
Recurrence: (n-2)*a(n) = n*(2*n-5)*a(n-1) - (n-4)*(n-1)*n*a(n-2). - Vaclav Kotesovec, Mar 20 2023

A212559 Number of functions f:{1,2,...,n}->{1,2,...,n} such that every non-recurrent element has at most one preimage.

Original entry on oeis.org

1, 1, 4, 27, 244, 2745, 36966, 580111, 10399096, 209672721, 4696872490, 115732052271, 3110867569140, 90587751885241, 2840805169411678, 95450112571474095, 3420897993621996016, 130266500391456691233, 5252293203395848789842, 223535386151669737094095, 10014286301754519472897900

Offset: 0

Geoffrey Critzer, May 21 2012

nonn

An element x of {1,2,...,n} is a recurrent element if there exists a positive integer k such that (f^k)(x) = x where f^k is the k-th iteration of functional composition.

The functional digraphs are composed of cycles of rooted trees with every non-root vertex of degree 1 or 2. Cf. A006152.

nn=20;a=x Exp[x/(1-x)];Range[0,nn]! CoefficientList[Series[1/(1-a),{x,0,nn}],x]

E.g.f.: 1/(1-A(x)) where A(x) is the e.g.f. for A006152.

cp's OEIS Frontend

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

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A006154 Number of labeled ordered partitions of an n-set into odd parts.

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A006155 Expansion of e.g.f.: 1/(2-x-exp(x)).

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A206703 Triangular array read by rows. T(n,k) is the number of partial permutations (injective partial functions) of {1,2,...,n} that have exactly k elements in a cycle. The k elements are not necessarily in the same cycle. A fixed point is considered to be in a cycle.

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A320264 Number T(n,k) of proper multisets of nonempty words with a total of n letters over k-ary alphabet such that all k letters occur at least once in the multiset; triangle T(n,k), n>=2, 1<=k<=n-1, read by rows.

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A114329 Triangle T(n,k) is the number of partitions of an n-set into lists (cf. A000262) with k lists of size 1.

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