A003575 -id:A003575 - cp's OEIS frontend

A007405 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=2.

1, 2, 6, 24, 116, 648, 4088, 28640, 219920, 1832224, 16430176, 157554048, 1606879040, 17350255744, 197553645440, 2363935624704, 29638547505408, 388328781668864, 5304452565517824, 75381218537805824, 1112348880749130752, 17014743624340539392, 269360902955086379008

Offset: 0

N. J. A. Sloane

Binomial transform of A004211.
Equals leftmost term in iterates of M^n * [1,1,1,...], where M = a bidiagonal matrix with (1,3,5,7,...) in the main diagonal and (1,1,1,...) in the superdiagonal. - Gary W. Adamson, Apr 13 2009
This is the number of type B set partitions, see R. Suter. - Per W. Alexandersson, Dec 19 2022

			a(4) = 116 = sum of top row terms of M^3 = (49 + 44 + 18 + 4 + 1).

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
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
Hasan Arslan, Nazmiye Alemdar, Mariam Zaarour, and Hüseyin Altındiş, On Bell numbers of type D, arXiv:2504.16522 [math.CO], 2025. See p. 3.
Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, Flattened Catalan Words, arXiv:2405.05357 [math.CO], 2024. See p. 2.
Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.
Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See p. 25.
Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, Flattened Stirling Permutations, arXiv:2306.13034 [math.CO], 2023. See p. 14.
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.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
John M. Neuberger, Nándor Sieben, and James W. Swift, Invariant Polydiagonal Subspaces of Matrices and Constraint Programming, arXiv:2411.10904 [math.DS], 2024. See p. 7.
Tilman Piesk, Sets of fixed points of permutations of the n-cube: a(3)=24 for the cube and a(4)=116 for the tesseract.
N. J. A. Sloane, Transforms
R. Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.

Cf. A000110 (b=1), this sequence (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

Cf. A004211, A039755, A039756, A187251.

m:=20; c:=2; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019

max = 19; f[x_]:= Exp[x + Exp[2x]/2 -1/2]; CoefficientList[Series[f[x], {x,0,max}], x]*Range[0, max]! (* Jean-François Alcover, Nov 22 2011 *)
Table[Sum[Binomial[n, k] * 2^k * BellB[k, 1/2], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

x='x+O('x^66); Vec(serlaplace(exp(x+1/2*exp(2*x)-1/2))) \\ Joerg Arndt, May 13 2013

@CachedFunction
def S(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return S(n-1, k-1, m) + (m*(k+1)-1)*S(n-1, k, m)
def A007405(n): return add(S(n, k, 2) for k in (0..n)) # Peter Luschny, May 20 2013

b=2;
def A007405_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(x +(exp(b*x)-1)/b) ).egf_to_ogf().list()
A007405_list(30) # G. C. Greubel, Feb 24 2019

E.g.f.: exp(x + (exp(2*x) - 1)/2).
Row sums of triangles A039755, A039756. - Philippe Deléham, Feb 20 2005
a(n) = sum of top row terms of M^n, M = an infinite square production matrix in which a diagonal of 1's is appended to the right of Pascal's triangle squared; as follows:
   1,  1,  0, 0, 0, 0, ...
   2,  1,  1, 0, 0, 0, ...
   4,  4,  1, 1, 0, 0, ...
   8, 12,  6, 1, 1, 0, ...
  16, 32, 24, 8, 1, 1, ...
   ... - Gary W. Adamson, Aug 01 2011
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-(2*k+1)*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
G.f.: -G(0) where G(k) = 1 - (x*(2*k+1) - 2)/(x*(2*k+1) - 1 - x*(x*(2*k+1) - 1)/(x + (x*(2*k+1) - 2)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 29 2013
G.f.: 1/Q(0), where Q(k) = 1 - 2*(k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: 1/Q(0), where Q(k) = 1 - x - x/(1 - x*(2*k+2)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 13 2013
G.f.: 1/(1-x*Q(0)), where Q(k) = 1 + x/(1 - x + 2*x*(k+1)/(x - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
Conjecture: Let M_n be an n X n matrix whose elements are m_ij = 1 for i < j - 1, m_ij = -1 for i = j - 1, and m_ij = binomial(n - i,j - i) otherwise. Then a(n - 1) = Det(M_n). - Benedict W. J. Irwin, Apr 19 2017
a(n) = exp(-1/2) * Sum_{k>=0} (2*k + 1)^n / (2^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) = Sum_{k=0..n} binomial(n,k) * A187251(k) * A187251(n-k). - Vaclav Kotesovec, Apr 17 2020
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

Name edited by G. C. Greubel, Feb 24 2019

A003576 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=4.

Original entry on oeis.org

1, 2, 8, 48, 352, 3008, 29440, 324096, 3947520, 52541440, 757260288, 11733385216, 194272854016, 3419584921600, 63707979972608, 1251489089060864, 25836869372608512, 558946705406427136, 12638569755079344128, 298003073694026432512, 7312035980392431353856

Offset: 0

N. J. A. Sloane

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..230
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), this sequence (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

m:=20; c:=4; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x+(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Feb 22 2019

seq(coeff(series(factorial(n)*exp(z+(1/4)*exp(4*z)-(1/4)),z,n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 22 2019

With[{m=20, b=4}, CoefficientList[Series[Exp[x+(Exp[b*x]-1)/b], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 22 2019 *)
Table[Sum[Binomial[n, k] * 4^k * BellB[k, 1/4], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

my(x='x+O('x^20)); b=4; Vec(serlaplace(exp(x+(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 22 2019

m = 20; b=4; T = taylor(exp(x+(exp(b*x)-1)/b), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 22 2019

E.g.f.: exp(z + (exp(4*z) - 1)/4).
G.f.: 1/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - 2*x^2*(2*k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 26 2013
a(n) = exp(-1/4) * Sum_{k>=0} (4*k + 1)^n / (4^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 4^(n + 1/4) * n^(n + 1/4) * exp(n/LambertW(4*n) - n - 1/4) / (sqrt(1 + LambertW(4*n)) * LambertW(4*n)^(n + 1/4)). - Vaclav Kotesovec, Jun 26 2022

A003577 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=5.

Original entry on oeis.org

1, 2, 9, 63, 536, 5307, 60389, 775988, 11062391, 172638727, 2921519374, 53221709973, 1037320865141, 21517178350762, 472862758184789, 10966587174511443, 267502464814857936, 6842498829509972687, 183057455239626138009, 5110016898453125496548

Offset: 0

N. J. A. Sloane

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..440
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), this sequence (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

b:=5;; a:=[1,2];; for n in [3..20] do a[n]:=2*a[n-1]+Sum([0..n-3],i->Binomial(n-2,i)*b^(n-2-i)*a[i+1]); od; Print(a); # Muniru A Asiru, Apr 10 2019

m:=20; c:=5; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019

With[{m=20, b=5}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b],{x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 24 2019 *)
Table[Sum[Binomial[n, k] * 5^k * BellB[k, 1/5], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

my(x='x+O('x^20)); b=5; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019

m = 20; b=5; T = taylor(exp(x + (exp(b*x) -1)/b), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 24 2019

E.g.f.: exp(x + (exp(5*x) - 1)/5).
a(n) = exp(-1/5) * Sum_{k>=0} (5*k + 1)^n / (5^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 5^(n + 1/5) * n^(n + 1/5) * exp(n/LambertW(5*n) - n - 1/5) / (sqrt(1 + LambertW(5*n)) * LambertW(5*n)^(n + 1/5)). - Vaclav Kotesovec, Jun 26 2022

Name clarified by Muniru A Asiru, Feb 24 2019

A003578 Dowling numbers: e.g.f. exp(x + (exp(b*x) - 1)/b), with b=6.

Original entry on oeis.org

1, 2, 10, 80, 772, 8648, 111592, 1631360, 26518672, 472528160, 9139219360, 190461416192, 4250569655872, 101040920561792, 2546488866632320, 67772341398044672, 1898177372174512384, 55780954727160472064, 1715291443214323558912, 55062161002484359565312

Offset: 0

N. J. A. Sloane

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..180
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), this sequence (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

m:=20; c:=6; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019

seq(coeff(series(factorial(n)*exp(z+(1/6)*exp(6*z)-(1/6)),z,n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 23 2019

With[{nn=20},CoefficientList[Series[Exp[x+Exp[6x]/6-1/6],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 12 2017 *)
Table[Sum[Binomial[n, k] * 6^k * BellB[k, 1/6], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

my(x='x+O('x^20)); b=6; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019

m = 20; b=6; T = taylor(exp(x + (exp(b*x) -1)/b), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 24 2019

E.g.f.: exp(x + (exp(6*x) - 1)/6).
a(n) = exp(-1/6) * Sum_{k>=0} (6*k + 1)^n / (6^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 6^(n + 1/6) * n^(n + 1/6) * exp(n/LambertW(6*n) - n - 1/6) / (sqrt(1 + LambertW(6*n)) * LambertW(6*n)^(n + 1/6)). - Vaclav Kotesovec, Jun 26 2022

A003580 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=8.

Original entry on oeis.org

1, 2, 12, 120, 1424, 19488, 307904, 5539712, 111259904, 2454487552, 58847153152, 1522019629056, 42209521995776, 1248370355347456, 39186678731423744, 1300179383923212288, 45436201241711542272, 1667242078056889843712, 64063345344029286727680

Offset: 0

N. J. A. Sloane

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..190
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.
Prudence Djagba and Jan Hązła, Combinatorics of subgroups of Beidleman near-vector spaces, arXiv:2306.16421 [math.RA], 2023.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), this sequence (b=8), A003581 (b=9), A003582 (b=10).

m:=20; c:=8; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019

seq(coeff(series(factorial(n)*exp(z+(1/8)*exp(8*z)-(1/8)),z,n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 24 2019

With[{m=20, b=8}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b],{x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 24 2019 *)
Table[Sum[Binomial[n, k] * 8^k * BellB[k, 1/8], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

my(x='x+O('x^20)); b=8; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019

m = 20; b=8; T = taylor(exp(x + (exp(b*x) -1)/b), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 24 2019

E.g.f.: exp(x + (exp(8*x) - 1)/8).
a(n) = exp(-1/8) * Sum_{k>=0} (8*k + 1)^n / (8^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 8^(n + 1/8) * n^(n + 1/8) * exp(n/LambertW(8*n) - n - 1/8) / (sqrt(1 + LambertW(8*n)) * LambertW(8*n)^(n + 1/8)). - Vaclav Kotesovec, Jun 26 2022

Name clarified by Muniru A Asiru, Feb 24 2019

A003579 Dowling numbers: e.g.f. exp(x + (exp(b*x) - 1)/b), with b=7.

Original entry on oeis.org

1, 2, 11, 99, 1066, 13283, 190933, 3117900, 56729565, 1132679479, 24564972756, 574431351673, 14394977015245, 384489778509034, 10894501505088695, 326149933663962479, 10280153573323314858

Offset: 0

N. J. A. Sloane

nonn

Muniru A Asiru, Table of n, a(n) for n = 0..425
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), this sequence (b=7), A003580 (b=8), A003581 (b=9), A003582 (b=10).

b:=7;; a:=[1,2];; for n in [3..20] do a[n]:=2*a[n-1]+Sum([0..n-3],i->Binomial(n-2,i)*b^(n-2-i)*a[i+1]); od; Print(a); # Muniru A Asiru, Apr 10 2019

m:=20; c:=7; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019

With[{m=20, b=7}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b],{x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 24 2019 *)
Table[Sum[Binomial[n, k] * 7^k * BellB[k, 1/7], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

Vec(serlaplace( exp(z + 1/7 * exp(7 * z) - 1/7) ) ) \\ Joerg Arndt, Feb 24 2019

m = 20; b=7; T = taylor(exp(x + (exp(b*x) -1)/b), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 24 2019

E.g.f.: exp(x + (exp(7*x) - 1)/7).
a(n) = exp(-1/7) * Sum_{k>=0} (7*k + 1)^n / (7^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 7^(n + 1/7) * n^(n + 1/7) * exp(n/LambertW(7*n) - n - 1/7) / (sqrt(1 + LambertW(7*n)) * LambertW(7*n)^(n + 1/7)). - Vaclav Kotesovec, Jun 26 2022

Name clarified by Muniru A Asiru, Feb 24 2019

A003581 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=9.

Original entry on oeis.org

1, 2, 13, 143, 1852, 27563, 473725, 9290396, 203745235, 4912490375, 128777672338, 3643086083981, 110557605978901, 3579776914324250, 123074955978249433, 4474133111905169219, 171363047274358839412, 6893620459732188296591, 290475101469031118494993

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 2*x + 13*x^2 + 143*x^3 + 1852*x^4 + 27563*x^5 + ...

Muniru A Asiru, Table of n, a(n) for n = 0..180
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), this sequence (b=9), A003582 (b=10), A364069 (b=63), A364070 (b=624).

m:=20; c:=9; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019

seq(coeff(series(factorial(n)*exp(z+(1/9)*exp(9*z)-(1/9)),z,n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 24 2019

With[{m=20, b=9}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b],{x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 24 2019 *)
Table[Sum[Binomial[n, k] * 9^k * BellB[k, 1/9], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

Vec(serlaplace(exp(z + (exp(9*z) - 1)/9))) \\ Michel Marcus, Nov 07 2014

m = 20; b=9; T = taylor(exp(x +(exp(b*x)-1)/b), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 24 2019

E.g.f.: exp(x + (exp(9*x) - 1)/9).
G.f.: 1/W(0), where W(k) = 1 - x - x/(1 - 9*(k+1)*x/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 07 2014
a(n) = exp(-1/9) * Sum_{k>=0} (9*k + 1)^n / (9^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 9^(n + 1/9) * n^(n + 1/9) * exp(n/LambertW(9*n) - n - 1/9) / (sqrt(1 + LambertW(9*n)) * LambertW(9*n)^(n + 1/9)). - Vaclav Kotesovec, Jun 26 2022

Name clarified by Muniru A Asiru, Feb 24 2019

A003582 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=10.

Original entry on oeis.org

1, 2, 14, 168, 2356, 37832, 701464, 14866848, 352943376, 9219925792, 261954304224, 8033968939648, 264411579439936, 9288709762556032, 346608927301622144, 13680000261825018368, 569006722158124974336, 24864267879086770135552, 1138321277772163220033024

Offset: 0

N. J. A. Sloane

nonn

In general, for b > 0, if e.g.f. = exp(x + (exp(b*x) - 1)/b), then a(n) ~ b^(n + 1/b) * n^(n + 1/b) * exp(n/LambertW(b*n) - n - 1/b) / (sqrt(1 + LambertW(b*n)) * LambertW(b*n)^(n + 1/b)). - Vaclav Kotesovec, Jun 26 2022

Muniru A Asiru, Table of n, a(n) for n = 0..180
Moussa Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. 159 (1996), no. 1-3, 13-33.
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.

Cf. A000110 (b=1), A007405 (b=2), A003575 (b=3), A003576 (b=4), A003577 (b=5), A003578 (b=6), A003579 (b=7), A003580 (b=8), A003581 (b=9), this sequence (b=10).

m:=20; c:=10; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x +(Exp(c*x)-1)/c) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 24 2019

seq(coeff(series(factorial(n)*exp(z+(1/10)*exp(10*z)-(1/10)),z,n+1), z, n), n = 0 .. 20); # Muniru A Asiru, Feb 24 2019

With[{m=20, b=10}, CoefficientList[Series[Exp[x +(Exp[b*x]-1)/b],{x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 24 2019 *)
Table[Sum[Binomial[n, k] * 10^k * BellB[k, 1/10], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 17 2020 *)

my(x='x+O('x^20)); b=10; Vec(serlaplace(exp(x +(exp(b*x)-1)/b))) \\ G. C. Greubel, Feb 24 2019

m = 20; b=10; T = taylor(exp(x + (exp(b*x) -1)/b), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Feb 24 2019

E.g.f.: exp(x + (exp(10*x) - 1)/10).
a(n) = exp(-1/10) * Sum_{k>=0} (10*k + 1)^n / (10^k * k!). - Ilya Gutkovskiy, Apr 16 2020
a(n) ~ 10^(n + 1/10) * n^(n + 1/10) * exp(n/LambertW(10*n) - n - 1/10) / (sqrt(1 + LambertW(10*n)) * LambertW(10*n)^(n + 1/10)). - Vaclav Kotesovec, Jun 26 2022

Name clarified by Muniru A Asiru, Feb 24 2019

A337039 a(n) = exp(-1/3) * Sum_{k>=0} (3k - 1)^n / (3^k k!).

Original entry on oeis.org

1, 0, 3, 9, 54, 351, 2673, 22842, 216513, 2248965, 25351704, 307699965, 3995419365, 55207193328, 808078734999, 12480510487509, 202697232446070, 3451417004044323, 61450890989472837, 1141331486235356178, 22066085726516137149, 443236553318792110113, 9233934519951699602400

Offset: 0

Ilya Gutkovskiy, Aug 12 2020

nonn

Cf. A000296, A003575, A004212, A337038, A337040, A337041, A337042, A337043.

nmax = 22; CoefficientList[Series[Exp[(Exp[3 x] - 1)/3 - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k] 3^k a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[(-1)^(n - k) Binomial[n, k] 3^k BellB[k, 1/3], {k, 0, n}], {n, 0, 22}]

G.f. A(x) satisfies: A(x) = (1 - 3*x + x*A(x/(1 - 3*x))) / (1 - 2*x - 3*x^2).
G.f.: (1/(1 + x)) * Sum_{k>=0} (x/(1 + x))^k / Product_{j=1..k} (1 - 3*j*x/(1 + x)).
E.g.f.: exp((exp(3*x) - 1) / 3 - x).
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n-1,k) * 3^k * a(n-k-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A004212(k).
a(n) ~ 3^(n - 1/3) * n^(n - 1/3) * exp(n/LambertW(3*n) - n - 1/3) / (sqrt(1 + LambertW(3*n)) * LambertW(3*n)^(n - 1/3)). - Vaclav Kotesovec, Jun 26 2022

cp's OEIS Frontend

A007405 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=2.

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A003576 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=4.

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A003577 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=5.

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A003578 Dowling numbers: e.g.f. exp(x + (exp(b*x) - 1)/b), with b=6.

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A003580 Dowling numbers: e.g.f. exp(x + (exp(b*x)-1)/b) with b=8.

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A003579 Dowling numbers: e.g.f. exp(x + (exp(b*x) - 1)/b), with b=7.

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A337039 a(n) = exp(-1/3) * Sum_{k>=0} (3k - 1)^n / (3^k k!).

A355111 Expansion of e.g.f. 3 / (4 - 3x - exp(3x)).