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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

Examples

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

References

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

Links

• 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.

Crossrefs

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.

Programs

Magma

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

Mathematica

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 *)

PARI

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

Sage

@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

Sage

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

Formula

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

Extensions

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