A004211 -id:A004211 - cp's OEIS frontend

A055882 a(n) = 2^nBell(n). E.g.f.: exp(exp(2x)-1).

1, 2, 8, 40, 240, 1664, 12992, 112256, 1059840, 10827264, 118758400, 1389711360, 17258893312, 226463227904, 3127694491648, 45316785602560, 686826595745792, 10861264214949888, 178802342273744896, 3058036745204924416, 54236710945813430272, 995874184692762673152

Offset: 0

Christian G. Bower, Jun 09 2000

a(n) is the number of set partitions of {1,2,...,n} with a (possibly empty) subset of designated elements in each block. - Geoffrey Critzer, Sep 16 2012

Chai Wah Wu, Table of n, a(n) for n = 0..200

Cf. A000079, A000110, A055883, A143405.

[2^n*Bell(n): n in [0..20]]; // Vincenzo Librandi, Sep 19 2014

seq(add(binomial(n, k)*(bell(n)), k=0..n), n=0..18); # Zerinvary Lajos, Dec 01 2006
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j) *binomial(n-1, j-1)*2^j, j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 04 2019

nn=20;a=Exp[2x]-1;Range[0,nn]!CoefficientList[Series[Exp[a],{x,0,nn}],x]  (* Geoffrey Critzer, Sep 16 2012 *)
Table[2^n BellB[n], {n, 0, 20}] (* Vincenzo Librandi, Sep 19 2014 *)

# Python 3.2 or higher required
from itertools import accumulate
A055882_list, blist, b, n2 = [1,2], [1], 1, 4
for _ in range(2, 201):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A055882_list.append(b*n2)
    n2 *= 2 # Chai Wah Wu, Sep 19 2014

a(n) = exp(-1)*2^n*Sum_{k>=0} k^n/k!. - Benoit Cloitre, May 20 2002
G.f.: 1/(1-2*x/(1-2*x/(1-2*x/(1-4*x/(1-2*x/(1-6*x/(1-2*x/(1-8*x/(1-... (continued fraction). - Paul Barry, Oct 11 2009
G.f.: 1/(U(0) - 2*x) where U(k) = 1 + 2*x - 2*x*(k+1)/(1 - 2*x/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 12 2012
G.f.: G(0)/(1+2*x) where G(k) = 1 - 4*x*(k+1)/((2*k+1)*(4*x*k-1) - 2*x*(2*k+1)*(2*k+3)*(4*x*k-1)/(2*x*(2*k+3) - 2*(k+1)*(4*x*k+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
G.f.: G(0)/2 where G(k) = 1 - (2*x*k + 1)/(2*x*k - 1 - 2*x*(2*x*k - 1)/(2*x + (2*x*k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 30 2013
G.f.: 1/Q(0), where Q(k) = 1 - 2*(k+1)*x - 4*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
a(n) = Sum_{k=0..n} binomial(n,k) * A004211(k) * A004211(n-k). - Vaclav Kotesovec, Apr 17 2020

A009235 E.g.f. exp( sinh(x) / exp(x) ) = exp( (1-exp(-2*x))/2 ).

Original entry on oeis.org

1, 1, -1, -1, 9, -23, -25, 583, -3087, 4401, 79087, -902097, 4783801, 2361049, -348382697, 4102879415, -24288551071, -47413121055, 3214104039007, -44472852461857, 326386562502889, 417716032223049, -55104307651136313, 962111031220099495

Offset: 0

R. H. Hardin

Hankel transform is (-1)^binomial(n+1,2)*A108400. - Paul Barry, Apr 15 2010

Seiichi Manyama, Table of n, a(n) for n = 0..529
I. M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121 [math.CO], 2001.

Cf. A004211, A317996, A318179, A318180, A318181.

a := n -> (-2)^n*add(Stirling2(n,k)*(-1/2)^k, k=0..n):
seq(a(n), n=0..23); # Peter Luschny, Jan 06 2020

With[{nn=30},CoefficientList[Series[Exp[Sinh[x]/Exp[x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Jan 07 2013 *)
Table[(-2)^n BellB[n, -1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

x='x+O('x^66); /* that many terms */
v=Vec(serlaplace(exp(sinh(x)/exp(x)))) /* Joerg Arndt, May 19 2012 */

a(n) = Sum_{k=0..n} (-2)^(n-k)*Stirling2(n, k). - Vladeta Jovovic, Apr 04 2003
From Peter Bala, May 16 2012: (Start)
Recurrence equation: a(n+1) = Sum_{k = 0..n} (-2)^(n-k)*C(n,k)*a(k). Written umbrally this is a(n+1) = (a-2)^n (expand the binomial and replace a^k with a(k)). More generally, a*f(a) = f(a-2) holds umbrally for any polynomial f(x). An inductive argument then establishes the umbral recurrence a*(a+2)*(a+4)*...*(a+2*(n-1)) = 1 with a(0) = 1. Cf. A004211.
Touchard's congruence holds for odd prime p: a(p+k) = (a(k) + a(k+1)) (mod p) for k = 0,1,2, ... (adapt the proof of Theorem 10.1 in Gessel). In particular, a(p) = 2 (mod p) for odd prime p. (End)
From Sergei N. Gladkovskii, Sep 21 2012 - Oct 24 2013: (Start)
Continued fractions:
G.f.: (1/E(0)-1)/x where E(k)=  1 - x/(1 - 2*x + 2*x*(k+1)/E(k+1));
G.f.: 1 +x/G(0) where G(k)= 1 + 2*x/(1 + 1/(1 + 4*x*(k+1)/G(k+1)));
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+x*2*k)/(1-x/(x-1/G(k+1)));
G.f.: 1/Q(0) where Q(k)= 1 - x/(1 + 2*x*(k+1)/Q(k+1) );
G.f.: Q(0)/(1-x), where Q(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) + (1-x+2*x*k)*(1+x+2*x*k)/Q(k+1)). (End)
Lim sup n->infinity (abs(a(n))/n!)^(1/n) / (2*abs(exp(1/LambertW(-2*n)) / LambertW(-2*n))) = 1. - Vaclav Kotesovec, Aug 04 2014
a(n) = (-2)^n*B_n(-1/2), where B_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Oct 20 2015
G.f. A(x) satisfies: A(x) = 1 + x*A(x/(1 + 2*x))/(1 + 2*x). - Ilya Gutkovskiy, May 02 2019

Extended with signs by Olivier Gérard, Mar 15 1997

A075497 Stirling2 triangle with scaled diagonals (powers of 2).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 28, 12, 1, 16, 120, 100, 20, 1, 32, 496, 720, 260, 30, 1, 64, 2016, 4816, 2800, 560, 42, 1, 128, 8128, 30912, 27216, 8400, 1064, 56, 1, 256, 32640, 193600, 248640, 111216, 21168, 1848, 72, 1

Offset: 1

Wolfdieter Lang, Oct 02 2002

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(2*z) - 1)*x/2) - 1.
Subtriangle of (0, 2, 0, 4, 0, 6, 0, 8, 0, 10, 0, 12, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 13 2013
Also the inverse Bell transform of the double factorial of even numbers Product_ {k=0..n-1} (2*k+2) (A000165). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015
This is the exponential Riordan array [exp(2*x), (exp(2*x) - 1)/2] belonging to the derivative subgroup of the exponential Riordan group. In the notation of Corcino, this is the triangle of (2, 2)-Stirling numbers of the second kind. A factorization of the array as an infinite product is given in the example section. - Peter Bala, Feb 20 2025

			Triangle begins:
  [1];
  [2,1];
  [4,6,1]; p(3,x) = x*(4 + 6*x + x^2).
  ...;
Triangle (0, 2, 0, 4, 0, 6, 0, 8, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, ...) begins:
  1
  0,  1
  0,  2,   1
  0,  4,   6,   1
  0,  8,  28,  12,  1
  0, 16, 120, 100, 20, 1. - _Philippe Deléham_, Feb 13 2013
From _Peter Bala_, Feb 23 2025: (Start)
The array factorizes as
/ 1               \       /1             \ /1             \ /1            \
| 2    1           |     | 2   1          ||0  1           ||0  1          |
| 4    6   1       |  =  | 4   4   1      ||0  2   1       ||0  0  1       | ...
| 8   28  12   1   |     | 8  12   6  1   ||0  4   4  1    ||0  0  2  1    |
|16  120 100  20  1|     |16  32  24  8  1||0  8  12  6  1 ||0  0  4  4  1 |
|...               |     |...             ||...            ||...           |
where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 2*x), x/(1 - 2*x)) = P^2, where P denotes Pascal's triangle. See A038207. Cf. A143494. (End)

Alois P. Heinz, Rows n = 1..141, flattened
Peter Bala, The white diamond product of power series
Peter Bala, Factorising (r,b)-Stirling arrays
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv 1507.04803 [math.CO], 2015.
Roberto B. Corcino, The (r, β)-Stirling Numbers, The Mindanao Forum, Vol. XIV, No.2, pp. 91-99, 1999.
Roberto B. Corcino and Maribeth B. Montero, The (r, β)-Stirling Numbers in the Context of 0-1 Tableau, Jour. Math. Soc. of the Philippines, ISSN 0115-6926, Vol. 32, No. 1 (2009), pp. 45-52
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
Wolfdieter Lang, First 10 rows.
Toufik Mansour, Generalization of some identities involving the Fibonacci numbers, arXiv:math/0301157 [math.CO], 2003.
Emanuele Munarini, Characteristic, admittance and matching polynomials of an antiregular graph, Appl. Anal. Discrete Math 3 (1) (2009) 157-176.

Columns 1-7 are A000079, A006516, A016283, A025966, A075510-A075512.
Row sums are A004211.
Cf. A008277, A075498-A075505.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
       `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
        binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, Aug 13 2015
# Alternatively, giving the triangle in the form displayed in the Example section:
gf := exp(x*exp(z)*sinh(z)):
X := n -> series(gf, z, n+2):
Z := n -> n!*expand(simplify(coeff(X(n), z, n))):
A075497_row := n -> op(PolynomialTools:-CoefficientList(Z(n), x)):
seq(A075497_row(n), n=0..9); # Peter Luschny, Jan 14 2018

Table[(2^(n - m)) StirlingS2[n, m], {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)

for(n=1, 11, for(m=1, n, print1(2^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

# uses[inverse_bell_transform from A265605]
multifact_2_2 = lambda n: prod(2*k + 2 for k in (0..n-1))
inverse_bell_matrix(multifact_2_2, 9) # Peter Luschny, Dec 31 2015

a(n, m) = (2^(n-m)) * Stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*2)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 2*m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-2*k*x), m >= 1.
E.g.f. for m-th column: (((exp(2*x)-1)/2)^m)/m!, m >= 1.
The row polynomials in t are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+2*x)*d/dx. Cf. A008277. - Peter Bala, Nov 25 2011
From Peter Bala, Jan 13 2018: (Start)
n-th row polynomial R(n,x)= x o x o ... o x (n factors), where o is the deformed Hadamard product of power series defined in Bala, section 3.1.
R(n+1,x)/x = (x + 2) o (x + 2) o...o (x + 2) (n factors).
R(n+1,x) = x*Sum_{k = 0..n} binomial(n,k)*2^(n-k)*R(k,x).
Dobinski-type formulas: R(n,x) = exp(-x/2)*Sum_{i >= 0} (2*i)^n* (x/2)^i/i!; 1/x*R(n+1,x) = exp(-x/2)*Sum_{i >= 0} (2 + 2*i)^n* (x/2)^i/i!. (End)

A049376 Row sums of triangle A046089.

Original entry on oeis.org

1, 1, 4, 22, 154, 1306, 12976, 147484, 1883932, 26680924, 414468496, 7001104936, 127677078904, 2498712779512, 52209534323584, 1159559538626896, 27269218041047056, 676732851527182864, 17669429275516846912, 484087943980439097184, 13882791112964223876256

Offset: 0

Wolfdieter Lang

nonn

a(n) is the number of n-permutations where each cycle has two (not necessarily distinct) roots. Here a root means a designated element in a cycle. Cf. A000262 which gives the number of n-permutations with a single root in each cycle. Note that the order of designating the elements is not important. Cf. (A bijection from endofunctions to "doubly" rooted trees where the order of designating the roots is important) Miklos Bona, A Walk Through Combinatorics, World Scientific Publishing, 2006, page 216. - Geoffrey Critzer, May 17 2012.

			a(2) = 4 because we have: (1'')(2'');(1''2);(12'');(1'2') where the permutations are given in cycle notation and the two roots in each cycle are designated by a '.

Alois P. Heinz, Table of n, a(n) for n = 0..436
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Column k=3 of A291709.
Cf. A000085, A000262, A046089, A136658.
Cf. A004211, A375173.

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+1)!/2*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017
a := proc(n) option remember; `if`(n < 3, [1, 1, 4][n + 1],
a(n-1)*(3*n-2) - a(n-2)*3*(n-1)*(n-2) + a(n-3)*(n-1)*(n-2)*(n-3)) end:
seq(a(n), n=0..20); # after Emanuele Munarini, Peter Luschny, Sep 09 2017

nn = 15;Drop[Range[0, nn]! CoefficientList[Series[Exp[x/(1 - x) + x^2/2/(1 - x)^2], {x, 0, nn}], x], 1]  (* Geoffrey Critzer, May 17 2012 *)

E.g.f.: exp(p(x)) with p(x) := x*(2-x)/(2*(1-x)^2) (E.g.f. first column of A046089).
Lah transform of A000085: a(n) = Sum_{k=0..n} n!/k!*binomial(n-1,k-1) * A000085(k). - Vladeta Jovovic, Oct 02 2003
a(n+3) - (3*n+7)*a(n+2) + 3*(n+1)*(n+2)*a(n+1) - n*(n+1)*(n+2)* a(n) = 0. - Emanuele Munarini, Sep 08 2017
a(n) ~ n^(n-1/6) / sqrt(3) * exp(-1/3 + n^(1/3)/2 + 3*n^(2/3)/2 - n). - Vaclav Kotesovec, Oct 23 2017
E.g.f.: Sum_{n>=0} ( Integral 1/(1-x)^3 dx )^n / n!, where the constant of integration is taken to be zero. - Paul D. Hanna, Apr 27 2019
From Seiichi Manyama, Jan 18 2025: (Start)
a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A004211(k).
a(n) = (1/exp(1/2)) * (-1)^n * n! * Sum_{k>=0} binomial(-2*k,n)/(2^k * k!). (End)

a(0)=1 prepended by Alois P. Heinz, Aug 01 2017

A075509 Shifts one place left under 10th-order binomial transform.

Original entry on oeis.org

1, 1, 11, 131, 1761, 27601, 506651, 10674211, 251686881, 6524202561, 183991725451, 5605930566051, 183428104316161, 6409252239788881, 237948848526923611, 9346097294356706051, 386966245108218203201, 16836505067572362863361, 767645305770283165781131

Offset: 0

Wolfdieter Lang, Oct 02 2002

Previous name was: a(n) are row sums of triangle A075505 (for n>=1).

Shifts one place left under k-th order binomial transform, k=1..10: A000110, A004211, A004212, A004213, A005011, A005012, A075506, A075507, A075508, A075509.

seq(10^n*BellB(n, 1/10), n=0..18); # Peter Luschny, Oct 20 2015

Table[10^n BellB[n, 1/10], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

a(n) = Sum_{m=0..n} 10^(n-m)*S2(n,m) with S2(n,m) = A048993(n,m) (Stirling2).
E.g.f.: exp((exp(10*x)-1)/10).
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 10*j*x). - Ilya Gutkovskiy, Mar 21 2018
a(n) ~ 10^n * n^n * exp(n/LambertW(10*n) - 1/10 - n) / (sqrt(1 + LambertW(10*n)) * LambertW(10*n)^n). - Vaclav Kotesovec, Jul 15 2021

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015

A075506 Shifts one place left under 7th-order binomial transform.

Original entry on oeis.org

1, 1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, 235877034446341, 6634976621814472, 197269776623577659, 6177654735731310917, 203136983117907790890, 6994626418539177737803, 251584328242318030774781

Offset: 0

Wolfdieter Lang, Oct 02 2002

Previous name was: Row sums of triangle A075502 (for n>=1).

Muniru A Asiru, Table of n, a(n) for n = 0..110

Shifts one place left under k-th order binomial transform, k=1..10: A000110, A004211, A004212, A004213, A005011, A005012, A075506, A075507, A075508, A075509.

List([0..20],n->Sum([0..n],m->7^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018

[seq(factorial(k)*coeftayl(exp((exp(7*x)-1)/7), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018

Mathematica
```
Table[7^n BellB[n, 1/7], {n, 0, 20}]
```

a(n) = sum((7^(n-m))*S2(n,m), m=0..n), with S2(n,m) = A008277(n,m) (Stirling2).
E.g.f.: exp((exp(7*x)-1)/7).
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 7*j*x). - Ilya Gutkovskiy, Mar 20 2018
a(n) ~ 7^n * n^n * exp(n/LambertW(7*n) - 1/7 - n) / (sqrt(1 + LambertW(7*n)) * LambertW(7*n)^n). - Vaclav Kotesovec, Jul 15 2021

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015

A075507 Shifts one place left under 8th-order binomial transform.

Original entry on oeis.org

1, 1, 9, 89, 1009, 13457, 210105, 3747753, 74565473, 1628999841, 38704241897, 993034281593, 27340167242321, 803154583649329, 25050853217628313, 826165199464341705, 28707262835597618369, 1047731789671001235265, 40053733152627299592137, 1599910554128824794493593

Offset: 0

Wolfdieter Lang, Oct 02 2002

Previous name was: Row sums of triangle A075503 (for n>=1).

Muniru A Asiru, Table of n, a(n) for n = 0..110

Shifts one place left under k-th order binomial transform, k=1..10: A000110, A004211, A004212, A004213, A005011, A005012, A075506, A075507, A075508, A075509.

List([0..20],n->Sum([0..n],m->8^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018

[seq(factorial(k)*coeftayl(exp((exp(8*x)-1)/8), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018

Table[8^n BellB[n, 1/8], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

a(n) = Sum_{m=0..n} 8^(n-m)*S2(n,m), with S2(n,m) = A008277(n,m) (Stirling2).
E.g.f.: exp((exp(8*x)-1)/8).
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 8*j*x). - Ilya Gutkovskiy, Mar 20 2018
a(n) ~ 8^n * n^n * exp(n/LambertW(8*n) - 1/8 - n) / (sqrt(1 + LambertW(8*n)) * LambertW(8*n)^n). - Vaclav Kotesovec, Jul 15 2021

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015

A075508 Shifts one place left under 9th-order binomial transform.

Original entry on oeis.org

1, 1, 10, 109, 1351, 19612, 333451, 6493069, 141264820, 3376695763, 87799365343, 2465959810690, 74353064138749, 2393123710957813, 81812390963020066, 2958191064076428793, 112727516544416978299, 4513118224822056822772, 189305466502867876489519

Offset: 0

Wolfdieter Lang, Oct 02 2002

Previous name was: Row sums of triangle A075504 (for n>=1).

Muniru A Asiru, Table of n, a(n) for n = 0..108

Shifts one place left under k-th order binomial transform, k=1..10: A000110, A004211, A004212, A004213, A005011, A005012, A075506, A075507, A075508, A075509.

List([0..20],n->Sum([0..n],m->9^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018

[seq(factorial(k)*coeftayl(exp((exp(9*x)-1)/9), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018

Table[9^n BellB[n, 1/9], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

a(n) = Sum_{m=0..n} 9^(n-m)*S2(n,m), with S2(n,m) = A008277(n,m) (Stirling2).
E.g.f.: exp((exp(9*x)-1)/9).
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 9*j*x). - Ilya Gutkovskiy, Mar 20 2018
a(n) ~ 9^n * n^n * exp(n/LambertW(9*n) - 1/9 - n) / (sqrt(1 + LambertW(9*n)) * LambertW(9*n)^n). - Vaclav Kotesovec, Jul 15 2021

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015

A301419 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - njx).

Original entry on oeis.org

1, 1, 3, 19, 201, 3176, 69823, 2026249, 74565473, 3376695763, 183991725451, 11854772145800, 890415496931689, 77023751991841669, 7592990698770559111, 845240026276785888451, 105409073489605774592897, 14625467507717709778793020, 2244123413703647502288608467, 378751257186051653931253015229

Offset: 0

Ilya Gutkovskiy, Mar 20 2018

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..101
N. J. A. Sloane, Transforms

Cf. A000110, A004211, A004212, A004213, A005011, A005012, A008277, A075506, A075507, A075508, A075509, A242817, A292914, A318183.

List([0..20],n->Sum([0..n],k->n^(n-k)*Stirling2(n,k))); # Muniru A Asiru, Mar 20 2018

Table[SeriesCoefficient[Sum[x^k/Product[(1 - n j x), {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[n! SeriesCoefficient[Exp[(Exp[n x] - 1)/n], {x, 0, n}], {n, 19}]]
Join[{1}, Table[Sum[n^(n - k) StirlingS2[n, k], {k, 0, n}], {n, 19}]]
(* Or: *)
A301419[n_] := If[n == 0, 1, n^n BellB[n, 1/n]];
Table[A301419[n], {n, 0, 19}] (* Peter Luschny, Dec 22 2021 *)

a(n) = sum(k=0, n, n^(n-k)*stirling(n, k, 2)); \\ Michel Marcus, Mar 23 2018

a(n) = n! * [x^n] exp((exp(n*x) - 1)/n), for n > 0.
a(n) = Sum_{k=0..n} n^(n-k)*Stirling2(n,k).
a(n) = n^n * BellPolynomial(n, 1/n) for n >= 1. - Peter Luschny, Dec 22 2021
a(n) ~ exp(n/LambertW(n^2) - n) * n^(2*n) / (sqrt(1 + LambertW(n^2)) * LambertW(n^2)^n). - Vaclav Kotesovec, Jun 06 2022

A337038 a(n) = exp(-1/2) * Sum_{k>=0} (2k - 1)^n / (2^k k!).

Original entry on oeis.org

1, 0, 2, 4, 20, 96, 552, 3536, 25104, 194816, 1637408, 14792768, 142761280, 1464117760, 15886137984, 181667507456, 2182268117248, 27456279388160, 360872502280704, 4943580063237120, 70437638474568704, 1041911242274562048, 15972832382065977344, 253388070573020401664

Offset: 0

Ilya Gutkovskiy, Aug 12 2020

nonn

Robert Israel, Table of n, a(n) for n = 0..516

Cf. A000296, A004211, A007405, A124311, A166922, A217203, A337039, A337040, A337041, A337042, A337043.

E:= exp((exp(2*x)-1)/2-x):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Aug 26 2020

nmax = 23; CoefficientList[Series[Exp[(Exp[2 x] - 1)/2 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 2^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 2^k BellB[k, 1/2], {k, 0, n}], {n, 0, 23}]

G.f. A(x) satisfies: A(x) = (1 - 2*x + x*A(x/(1 - 2*x))) / (1 - x - 2*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 2*j*x/(1 + x)).
E.g.f.: exp((exp(2*x) - 1) / 2 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 2^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004211(k).
a(n) ~ 2^(n - 1/2) * n^(n - 1/2) * exp(n/LambertW(2*n) - n - 1/2) / (sqrt(1 + LambertW(2*n)) * LambertW(2*n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022

cp's OEIS Frontend

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A055882 a(n) = 2^nBell(n). E.g.f.: exp(exp(2x)-1).

A301419 a(n) = [x^n] Sum_{k>=0} x^k/Product_{j=1..k} (1 - njx).

A337038 a(n) = exp(-1/2) * Sum_{k>=0} (2k - 1)^n / (2^k k!).