A050351 -id:A050351 - cp's OEIS frontend

A257565 Generalized Fubini numbers. Square array read by ascending antidiagonals, A(n,k) = 1 + k(Sum_{j=1..n-1} C(n,j)A(j,k)); n>=0 and k>=0.

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 13, 5, 1, 1, 1, 75, 37, 7, 1, 1, 1, 541, 365, 73, 9, 1, 1, 1, 4683, 4501, 1015, 121, 11, 1, 1, 1, 47293, 66605, 17641, 2169, 181, 13, 1, 1, 1, 545835, 1149877, 367927, 48601, 3971, 253, 15, 1, 1, 1, 7087261, 22687565, 8952553, 1306809, 108901, 6565, 337, 17, 1, 1

Offset: 0

Peter Luschny, May 08 2015

M. Mureşan defined the generalized Fubini numbers as the enumerators of the k-labeled ordered p partitions of an n-set.

			      1,       1,       1,       1,        1,         1, ...  A000012
      1,       1,       1,       1,        1,         1, ...  A000012
      1,       3,       5,       7,        9,        11, ...  A005408
      1,      13,      37,      73,      121,       181, ...  A003154
      1,      75,     365,    1015,     2169,      3971, ...  A193252
      1,     541,    4501,   17641,    48601,    108901, ...
      1,    4683,   66605,  367927,  1306809,   3583811, ...
      1,   47293, 1149877, 8952553, 40994521, 137595781, ...
A000012, A000670, A050351, A050352,  A050353,

M. Mureşan, On the generalized Fubini numbers. (Romanian) Stud. Cercet. Mat. 37, 70-76 (1985).

Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.
N. Kilar and Y. Simsek, A new family of Fubini type numbers and polynomials associated with Apostol-Bernoulli numbers and polynomials, J. Korean Math. Soc. 54 (5) (2017) 1605-1621.

Rows: A005408, A003154, A193252.

Columns: A000670, A050351, A050352, A050353.

F := proc(n,k) option remember; 1+k*add(binomial(n,j)*F(j,k),j=1..n-1) end:
seq(print(seq(F(n-k,k),k=0..n)), n=0..7); # triangular form
egf := k -> 1+1/(1/(exp(z)-1)-k): # egf of column k
for k from 0 to 4 do seq(j!*coeff(series(egf(k),z,10),z,j),j=0..8) od;
A := (n,k) -> `if`(n=0,1,add(k^(n-j-1)*(k+1)^j*combinat:-eulerian1(n,j),j=0..n-1)): seq(print(seq(A(n,k),k=0..5)),n=0..7);

A[n_, k_] := A[n, k] = 1 + k Sum[Binomial[n, j] A[j, k], {j, 1, n - 1}]; Table[A[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 30 2016 *)

E.g.f. of column k: 1+1/(1/(exp(z)-1)-k).
A(n,k) = Sum_{j=0..n-1} k^j*j!*{n,j+1} for n>0, else 1; {n,j} denotes the Stirling subset numbers.
A(n,k) = Sum_{j=0..n-1} k^(n-j-1)*(k+1)^j* for n>0, else 1;  denotes the Eulerian numbers.

A321189 a(n) = n! * [x^n] 1 - 1/(n - 1/(exp(x) - 1)).

Original entry on oeis.org

1, 1, 5, 73, 2169, 108901, 8288293, 890380177, 128364028145, 23918924529901, 5595490598128221, 1605718043992482553, 554663179293965398825, 227038711419826844827381, 108674023653792712066606229, 60142879347501714200454327841, 38108071228342727619600464659425

Offset: 0

Ilya Gutkovskiy, Oct 29 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..200
Robert Gill, The number of elements in a generalized partition semilattice, Discrete mathematics 186.1-3 (1998): 125-134. See Example 1.

Main diagonal of A257565.

Cf. A000670, A050351, A050352, A050353, A094420.

Concatenation([1],List([1..16],n->Sum([1..n],k->Stirling2(n,k)*Factorial(k)*n^(k-1)))); # Muniru A Asiru, Oct 29 2018

seq(coeff(series(factorial(n)*(1-1/(n-1/(exp(x)-1))),x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 29 2018
# Or, using the recurrence of the Fubini polynomials:
F := proc(n) option remember; if n = 0 then return 1 fi;
expand(add(binomial(n, k)*F(n-k)*x, k = 1..n)) end:
a := n -> `if`(n=0, 1, subs(x = n, F(n)) / n):
seq(a(n), n = 0..16);  # Peter Luschny, May 21 2021

Table[n! SeriesCoefficient[1 - 1/(n - 1/(Exp[x] - 1)), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[StirlingS2[n, k] k! n^(k - 1), {k, n}], {n, 16}]]

{a(n) = if(n==0, 1, sum(k=0, n, k!*n^(k-1)*stirling(n, k, 2)))} \\ Seiichi Manyama, Jun 12 2020

a(0) = 1; a(n) = Sum_{k=1..n} Stirling2(n, k)*k!*n^(k-1).
a(n) = A257565(n, n).
From Vaclav Kotesovec, Oct 29 2018: (Start)
a(n) ~ exp(1/2) * n! * n^(n-1).
a(n) ~ sqrt(2*Pi) * n^(2*n - 1/2) / exp(n - 1/2). (End)
a(n) = F_{n}(n) / n for n >= 1, where F_{n}(x) is the Fubini polynomial. In other words: a(n) = A094420(n) / n for n >= 1. - Peter Luschny, May 21 2021

A050356 Number of ordered factorizations of n with 2 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 4, 1, 7, 1, 16, 4, 7, 1, 40, 1, 7, 7, 64, 1, 40, 1, 40, 7, 7, 1, 208, 4, 7, 16, 40, 1, 73, 1, 256, 7, 7, 7, 292, 1, 7, 7, 208, 1, 73, 1, 40, 40, 7, 1, 1024, 4, 40, 7, 40, 1, 208, 7, 208, 7, 7, 1, 544, 1, 7, 40, 1024, 7, 73, 1, 40, 7, 73, 1, 1840, 1, 7, 40, 40, 7, 73, 1

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			For n=6 we have ((6)) = ((3*2)) = ((2*3)) = ((3)*(2)) = ((2)*(3)) = ((3))*((2)) = ((2))*((3)), thus a(6) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A002033, A050351, A050352, A050353, A050354, A050355, A050357, A050358, A050359.

A050356aux(n) = if(1==n,1/3, 3*sumdiv(n,d, if(dA050356aux(d), 0)));
A050356(n) = if(1==n,n,A050356aux(n)); \\ Antti Karttunen, May 19 2017, after the general recurrence given by Vladeta Jovovic May 25 2005 in A050354.

Dirichlet g.f.: (3-2*zeta(s))/(4-3*zeta(s)).
a(p^k) = 4^(k-1).
a(A002110(n)) = A050352(n).
Sum_{k=1..n} a(k) ~ -n^r / (9*r*Zeta'(r)), where r = 2.52138975790328306967497455387140053675965539610041801606891036... is the root of the equation Zeta(r) = 4/3. - Vaclav Kotesovec, Feb 02 2019

A050358 Number of ordered factorizations of n with 3 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 5, 1, 9, 1, 25, 5, 9, 1, 65, 1, 9, 9, 125, 1, 65, 1, 65, 9, 9, 1, 425, 5, 9, 25, 65, 1, 121, 1, 625, 9, 9, 9, 605, 1, 9, 9, 425, 1, 121, 1, 65, 65, 9, 1, 2625, 5, 65, 9, 65, 1, 425, 9, 425, 9, 9, 1, 1145, 1, 9, 65, 3125, 9, 121, 1, 65, 9, 121, 1, 4825, 1, 9, 65, 65, 9, 121

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

The Dirichlet inverse is given by A050356, turning all but the first element of A050356 negative. - R. J. Mathar, Jul 15 2010

			6 = (((6))) = (((3*2))) = (((2*3))) = (((3)*(2))) = (((2)*(3))) = (((3))*((2))) = (((2))*((3))) = (((3)))*(((2))) = (((2)))*(((3))).

R. J. Mathar, Table of n, a(n) for n = 1..899

Cf. A002033, A050351-A050359. a(p^k)=5^(k-1). a(A002110)=A050353.

Dirichlet g.f.: (4-3*zeta(s))/(5-4*zeta(s)).
a(n) = A050359(A101296(n)). - R. J. Mathar, May 26 2017
Sum_{k=1..n} a(k) ~ -n^r / (16*r*Zeta'(r)), where r = 2.7884327053324956670606046076818023223650950899573090550836329583345... is the root of the equation Zeta(r) = 5/4. - Vaclav Kotesovec, Feb 02 2019

A194649 Triangle of coefficients of a sequence of polynomials related to the enumeration of linear labeled rooted trees.

Original entry on oeis.org

1, 1, 3, 4, 13, 36, 24, 75, 316, 432, 192, 541, 3060, 6360, 5760, 1920, 4683, 33244, 92880, 127680, 86400, 23040, 47293, 403956, 1418424, 2620800, 2688000, 1451520, 322560, 545835, 5449756, 23051952, 53548992, 73785600, 60318720, 27095040, 5160960, 7087261, 80985780, 400813080, 1122145920, 1943867520, 2133734400

Offset: 0

Peter Bala, Sep 01 2011

Define the sequence of polynomials {P(n,x)}n>=0 recursively by setting P(0,x) = 1, P(1,x) = 1 and P(n+1,x) = d/dx((1+x)*(1+2*x)*P(n,x)) for n >= 1. The first few values are P(2,x) = 3 + 4*x, P(3,x) = 13 + 36*x + 24*x^2 and P(4,x) = 75 + 316*x + 432*x^2 + 192*x^3.
This triangle shows the coefficients of the P(n,x) in ascending powers of x. The values of P(n,x) at an integer or half-integer value of x enumerate linear labeled rooted trees: in particular we have P(n,0) = A000670(n), P(n,1/2) = A050351(n), P(n,1) = A050352(n) and P(n,3/2) = A050353(n).
More generally, for m >= 2, P(n,m/2-1), n = 0,1,2,... counts m level linear labeled rooted trees (see the e.g.f. below and the comment of Benoit Cloitre in A050351).

			Triangle begins
n\k|......0.......1........2........3........4........5.......6
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
.0.|......1
.1.|......1
.2.|......3.......4
.3.|.....13......36.......24
.4.|.....75.....316......432......192
.5.|....541....3060.....6360.....5760.....1920
.6.|...4683...33244....92880...127680....86400....23040
.7.|..47293..403956..1418424..2620800..2688000..1451520..322560
..

L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.

Cf. A000670, A002866 (main diagonal), A050351, A050352, A050353, A083411 (1/4*column 1).

T[0, 0] = T[1, 0] = 1; T[n_, k_] /; 0 <= k <= n-1 := T[n, k] = (k+1)*(2* T[n-1, k-1] + 3*T[n-1, k] + T[n-1, k+1]); T[, ] = 0;
{1}~Join~Table[T[n, k], {n, 1, 9}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 13 2019 *)

T(n,k) = 2^k*Sum_{i = k+1..n} Stirling2(n,i)*i!*binomial(i-1,k).
Recurrence: T(n+1,k) = (k+1)*(2*T(n,k-1)+3*T(n,k)+T(n,k+1)).
E.g.f.: G(x,t) := 1 + (1-exp(t))/((2*x+1)*exp(t)-2*x-2) = Sum_{n>=0} P(n,x)*t^n/n! = 1 + t + (3 + 4*x)*t^2/2! + (13 + 36*x + 24*x^2)*t^3/3! + ....
Column k generating function: 2^k*((exp(x)-1)/(2-exp(x)))^(k+1) (apart from initial term 1 when k = 0).
The generating function G(x,t) satisfies the partial differential equation d/dx((1+x)*(1+2*x)*G(x,t)) - d/dt(G(x,t)) = 2*(2x+1). Hence the row polynomials P(n,x) satisfy the defining recurrence P(n+1,x) = d/dx((1+x)*(2+x)*P(n,x)), with P(0,x) = P(1,x) = 1.
Reflection property: P(n,x) = (-1)^n*P(n,-x-3/2).
The polynomial P(n,x) has all real zeros, lying in the interval [-1,-1/2] (apply [Liu et al, Theorem 1.1, Corollary 1.2] with f(x) = P(n,x-1/2) and g(x) = P'(n,x-1/2) and use the reflection property).
Row sums are A050352; Column 0: A000670; Column 1: 4*A083411; Main diagonal: A002866.

A195205 Triangle of coefficients of a sequence of binomial type polynomials.

Original entry on oeis.org

3, 6, 9, 30, 54, 27, 222, 468, 324, 81, 2190, 5130, 4320, 1620, 243, 27006, 68400, 65610, 30780, 7290, 729, 399630, 1076166, 1135890, 618030, 187110, 30618, 2187, 6899262, 19532268, 22212792, 13471920, 4796820, 1020600, 122472, 6561

Offset: 1

Peter Bala, Sep 13 2011

Define a polynomial sequence P_n(x) by means of the recursion
P_(n+1)(x) = x*(P_n(x) + 2*P_n(x+1)), with P_0(x) = 1.
The first few values are
P_1(x) = 3*x, P_2(x) = 3*x*(3*x + 2),
P_3(x) = 3*x*(9*x^2 + 18*x + 10),
P_4(x) = 3*x*(27*x^3 + 108*x^2 + 156*x + 74).
The present table shows the coefficients of these polynomials (excluding P_0(x)) in ascending powers of x. Compare with A195204.
Triangle T(n,k) (1 <= k <= n), read by rows, given by (0, 2, 3, 4, 6, 6, 9, 8, 12, 10, 15, ...) DELTA (3, 0, 3, 0, 3, 0, 3, 0, 3, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011

			Triangle begins
n\k|.....1.......2......3......4......5......6
==============================================
..1|.....3
..2|.....6.......9
..3|....30......54.....27
..4|...222.....468....324.....81
..5|..2190....5130...4320...1620....243
..6|.27006...68400..65610..30780...7290....729
...
Triangle (0, 2, 3, 4, 6, 6, 9, ...) DELTA (3, 0, 3, 0, 3, 0, 3, 0, ...) begins:
1;
0,     3;
0,     6,     9;
0,    30,    54,    27;
0,   222,   468,   324,    81;
0,  2190,  5130,  4320,  1620,   243;
0, 27006, 68400, 65610, 30780,  7290,   729;
... - _Philippe Deléham_, Dec 22 2011

Wikipedia, Binomial type

Cf. A004123 ((1/3)*column 1), A050351 ((1/3)*row sums), A179929 (delta operator coeffs.), A195204.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n=0,3,polylog(-n, 2/3)), 10); # Peter Luschny, Jan 29 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[If[# == 0, 3, PolyLog[-#, 2/3]]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

E.g.f.: F(x,z) := (exp(z)/(3 - 2*exp(z)))^x = 1 + 3*x*z + (6*x + 9*x^2)*z^2/2! + (30*x + 54*x^2 + 27*x^3)*z^3/3! + ....
The generating function F(x,z) = Sum_{n>=0} P_n(x)*z^n/n! satisfies the partial differential equation d/dz(F(x,z)) = x*F(x,z) + 2*x*F(x+1,z). Hence the row generating polynomials P_n(x) satisfy the recurrence P_(n+1)(x) = x*(P_n(x) + 2*P_n(x+1)), with P_0(x) = 1. The form of the e.g.f. shows that the polynomials P_n(x) are a sequence of binomial type. In what follows we denote P_n(x) by x^[n].
Relation with rising factorials
x^[n] = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n,k)*3^k*x*(x+1)*...*(x+k-1),
and its inverse formula
3^n*x*(x+1)*...*(x+n-1) = Sum_{k=1..n} |Stirling1(n,k)|*x^[k].
The delta operator D*:
The row polynomials form a polynomial sequence of binomial type. If D denotes the derivative operator 1/3*d/dx then the associated delta operator D* is given by D* = D - 2*D^2/2! + 2*D^3/3! + 6*D^4/4! - 30*D^5/5! - ..., where the sequence of coefficients [1, -2, 2, 6, -30, -42, 882, ...] equals (-1)^n*A179929(n). D* is the lowering operator for the row polynomials, that is, (D*)x^[n] = n*x^[n-1].
Generalized Dobinski formula:
exp(-x)*Sum_{k >= 1} (-k)^[n]*x^k/k! = (-1)^n*Bell(n,3*x),
where the Bell (or exponential) polynomials are defined as
Bell(n,x) := Sum_{k = 1..n} Stirling2(n,k)*x^k.
Relation with the Bell polynomials:
The alternating n-th row entries (-1)^(n+k)*T(n,k) are the connection coefficients expressing the polynomial Bell(n,3*x) as a linear combination of Bell(k,x), 1 <= k <= n. For example for row 4:
Bell(4,3*x) = -222*Bell(1,x) + 468*Bell(2,x) - 324*Bell(3,x) + 81*Bell(4,x).
Generalized Bernoulli summation formula:
We have the following generalization of Bernoulli's formula for the sum of the powers of integers:
3*Sum_{k = 1..n} k^[p] = 1/(p+1)*Sum_{k = 0..p} (-1)^k * binomial(p+1,k)*B_k*n^[p+1-k], where B_k =[1, -1/2, 1/6, 0, -1/30, ...] denotes the sequence of Bernoulli numbers.
Relation with other sequences:
Row sums = 3*A050351(n) for n >= 1. Column 1 = 3*A004123.
T(n,k) = A185285(n,k)*3^k. - Philippe Deléham, Feb 17 2013
Also the Bell transform of 3*A004123. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016
Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 3*x*z/(1 - 2*z/(1 - 3*(x + 1)*z/(1 - 4*z/(1 - 3*(x + 2)*z/(1 - 6*z/(1 - 3*(x + 3)*z/(1 - 8*z/(1 - ... ))))))))). - Peter Bala, Dec 12 2024

A305988 Expansion of e.g.f. 1/(1 + log(2 - exp(x))).

Original entry on oeis.org

1, 1, 4, 24, 194, 1970, 24062, 343294, 5601122, 102847794, 2098766582, 47117285270, 1154031484586, 30622256174458, 875092190716382, 26794239236959806, 875110094707912562, 30367988674208286914, 1115822099409002188358, 43276913813553367194598, 1766830322476935945014330

Offset: 0

Ilya Gutkovskiy, Jun 15 2018

nonn

Stirling transform of A007840.

			1/(1 + log(2 - exp(x))) = 1 + x + 4*x^2/2! + 24*x^3/3! + 194*x^4/4! + 1970*x^5/5! + 24062*x^6/6! + ...

Robert Israel, Table of n, a(n) for n = 0..402
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A000629, A000670, A007840, A050351, A083355, A305323.

b:= proc(n) b(n):= n!*`if`(n=0, 1, add(b(k)/(k!*(n-k)), k=0..n-1)) end:
a:= n-> add(Stirling2(n, j)*b(j$2), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018

nmax = 20; CoefficientList[Series[1/(1 + Log[2 - Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[StirlingS2[n, k] Abs[StirlingS1[k, j]] j!, {j, 0, k}], {k, 0, n}], {n, 0, 20}]

a(n) = Sum_{k=0..n} Stirling2(n,k)*A007840(k).

a(n) ~ n! / ((2*exp(1) - 1) * (log(2 - exp(-1)))^(n+1)). - Vaclav Kotesovec, Jul 01 2018

A331611 E.g.f.: exp(1 / (2 - cosh(x)) - 1) (even powers only).

Original entry on oeis.org

1, 1, 10, 241, 10585, 732826, 73233205, 9955632961, 1764233731270, 394629336427021, 108652463882802505, 36084903957564392206, 14217903951354603567385, 6554505383225768210009041, 3493988190176442653240091010, 2131975894217009666242489287001

Offset: 0

Ilya Gutkovskiy, Jan 22 2020

nonn

Cf. A005046, A050351, A075729, A094088, A331608, A331612.

nmax = 15; Table[(CoefficientList[Series[Exp[1/(2 - Cosh[x]) - 1], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A094088[0] = 1; A094088[n_] := A094088[n] = Sum[Binomial[2 n, 2 k] A094088[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] A094088[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n-1,2*k-1) * A094088(k) * a(n-k).

a(n) ~ 2^(2*n + 1/4) * exp(1/(2*sqrt(3)*log(2 + sqrt(3))) - 2/3 + sqrt(8*n/log(2 + sqrt(3)))/3^(1/4) - 2*n) * n^(2*n - 1/4) / (3^(1/8) * log(2 + sqrt(3))^(2*n + 1/4)). - Vaclav Kotesovec, Jan 26 2020

A384344 Expansion of Product_{k>=1} (1 + kx)^((1/6) (2/3)^k).

Original entry on oeis.org

1, 1, -2, 10, -77, 787, -9972, 150552, -2637729, 52615903, -1177590290, 29228602546, -796945212035, 23681656958269, -761803800466856, 26376749702235900, -978091742247376932, 38674335439691203644, -1624351949069462807480, 72221688529265896447384

Offset: 0

Seiichi Manyama, May 26 2025

sign

Cf. A381890, A384343, A384345.

Cf. A050351, A090351.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, (-1)^(k-1)*sum(j=0, k, 2^(j-1)*j!*stirling(k, j, 2))*x^k/k)))

G.f. A(x) satisfies A(x) = (1+x)^(1/3) * A(x/(1+x))^(2/3).
G.f.: exp(Sum_{k>=1} (-1)^(k-1) * A050351(k) * x^k/k).
G.f.: 1/B(-x), where B(x) is the g.f. of A090351.

A308878 Expansion of e.g.f. (1 - log(1 + x))/(1 - 2*log(1 + x)).

Original entry on oeis.org

1, 1, 3, 14, 86, 664, 6136, 66240, 816672, 11331552, 174662304, 2961774144, 54785368128, 1097882522112, 23693117756928, 547844658441216, 13511950038494208, 354086653712228352, 9824794572366544896, 287752569360558907392, 8871374335098501292032

Offset: 0

Ilya Gutkovskiy, Jun 29 2019

nonn

Inverse Stirling transform of A002866.

Cf. A002866, A008275, A011782, A050351, A088501, A306042, A308877.

nmax = 20; CoefficientList[Series[(1 - Log[1 + x])/(1 - 2 Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] 2^(k - 1) k!, {k, 1, n}], {n, 1, 20}]]

a(0) = 1; a(n) = Sum_{k=1..n} Stirling1(n,k) * 2^(k-1) * k!.

a(n) ~ n! * exp(1/2) / (4 * (exp(1/2) - 1)^(n+1)). - Vaclav Kotesovec, Jun 29 2019

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