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

%I A007405 M1674 #140 Apr 30 2025 10:13:19
%S A007405 1,2,6,24,116,648,4088,28640,219920,1832224,16430176,157554048,
%T A007405 1606879040,17350255744,197553645440,2363935624704,29638547505408,
%U A007405 388328781668864,5304452565517824,75381218537805824,1112348880749130752,17014743624340539392,269360902955086379008
%N A007405 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=2.
%C A007405 Binomial transform of A004211.
%C A007405 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
%C A007405 This is the number of type B set partitions, see R. Suter. - _Per W. Alexandersson_, Dec 19 2022
%D A007405 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007405 T. D. Noe, <a href="/A007405/b007405.txt">Table of n, a(n) for n = 0..100</a>
%H A007405 V. E. Adler, <a href="http://arxiv.org/abs/1510.02900">Set partitions and integrable hierarchies</a>, arXiv:1510.02900 [nlin.SI], 2015.
%H A007405 Hasan Arslan, Nazmiye Alemdar, Mariam Zaarour, and Hüseyin Altındiş, <a href="https://arxiv.org/abs/2504.16522">On Bell numbers of type D</a>, arXiv:2504.16522 [math.CO], 2025. See p. 3.
%H A007405 Jean-Luc Baril, Pamela E. Harris, and José L. Ramírez, <a href="https://arxiv.org/abs/2405.05357">Flattened Catalan Words</a>, arXiv:2405.05357 [math.CO], 2024. See p. 2.
%H A007405 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry3/barry84r2.html">A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.
%H A007405 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry2/barry281.html">Constructing Exponential Riordan Arrays from Their A and Z Sequences</a>, Journal of Integer Sequences, 17 (2014), #14.2.6.
%H A007405 Paul Barry, <a href="https://arxiv.org/abs/1702.04007">Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays</a>, arXiv:1702.04007 [math.CO], 2017.
%H A007405 Moussa Benoumhani, <a href="http://dx.doi.org/10.1016/0012-365X(95)00095-E">On Whitney numbers of Dowling lattices</a>, Discrete Math. 159 (1996), no. 1-3, 13-33.
%H A007405 Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, <a href="https://arxiv.org/abs/2308.14183">Combinatorial Identities for Vacillating Tableaux</a>, arXiv:2308.14183 [math.CO], 2023. See p. 25.
%H A007405 Adam Buck, Jennifer Elder, Azia A. Figueroa, Pamela E. Harris, Kimberly Harry, and Anthony Simpson, <a href="https://arxiv.org/abs/2306.13034">Flattened Stirling Permutations</a>, arXiv:2306.13034 [math.CO], 2023. See p. 14.
%H A007405 Paweł Hitczenko, <a href="https://arxiv.org/abs/2403.03422">A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality</a>, arXiv:2403.03422 [math.CO], 2024. See p. 8.
%H A007405 Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.pdf">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%H A007405 John M. Neuberger, Nándor Sieben, and James W. Swift, <a href="https://arxiv.org/abs/2411.10904">Invariant Polydiagonal Subspaces of Matrices and Constraint Programming</a>, arXiv:2411.10904 [math.DS], 2024. See p. 7.
%H A007405 Tilman Piesk, Sets of fixed points of permutations of the n-cube: a(3)=24 <a href="https://commons.wikimedia.org/wiki/Category:Cube_subspaces_(image_set)">for the cube</a> and a(4)=116 <a href="https://commons.wikimedia.org/wiki/Category:Tesseract_subspaces_(image_set)">for the tesseract</a>.
%H A007405 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A007405 R. Suter, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/SUTER/sut1.html">Two analogues of a classical sequence</a>, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
%F A007405 E.g.f.: exp(x + (exp(2*x) - 1)/2).
%F A007405 Row sums of triangles A039755, A039756. - _Philippe Deléham_, Feb 20 2005
%F A007405 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:
%F A007405    1,  1,  0, 0, 0, 0, ...
%F A007405    2,  1,  1, 0, 0, 0, ...
%F A007405    4,  4,  1, 1, 0, 0, ...
%F A007405    8, 12,  6, 1, 1, 0, ...
%F A007405   16, 32, 24, 8, 1, 1, ...
%F A007405    ... - _Gary W. Adamson_, Aug 01 2011
%F A007405 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
%F A007405 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
%F A007405 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
%F A007405 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
%F A007405 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
%F A007405 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
%F A007405 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + 1)^n / (2^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020
%F A007405 a(n) = Sum_{k=0..n} binomial(n,k) * A187251(k) * A187251(n-k). - _Vaclav Kotesovec_, Apr 17 2020
%F A007405 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
%e A007405 a(4) = 116 = sum of top row terms of M^3 = (49 + 44 + 18 + 4 + 1).
%t A007405 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 *)
%t A007405 Table[Sum[Binomial[n, k] * 2^k * BellB[k, 1/2], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)
%o A007405 (PARI) x='x+O('x^66); Vec(serlaplace(exp(x+1/2*exp(2*x)-1/2))) \\ _Joerg Arndt_, May 13 2013
%o A007405 (Sage)
%o A007405 @CachedFunction
%o A007405 def S(n, k, m):
%o A007405     if k > n or k < 0 : return 0
%o A007405     if n == 0 and k == 0: return 1
%o A007405     return S(n-1, k-1, m) + (m*(k+1)-1)*S(n-1, k, m)
%o A007405 def A007405(n): return add(S(n, k, 2) for k in (0..n)) # _Peter Luschny_, May 20 2013
%o A007405 (Sage)
%o A007405 b=2;
%o A007405 def A007405_list(prec):
%o A007405     P.<x> = PowerSeriesRing(QQ, prec)
%o A007405     return P( exp(x +(exp(b*x)-1)/b) ).egf_to_ogf().list()
%o A007405 A007405_list(30) # _G. C. Greubel_, Feb 24 2019
%o A007405 (Magma) m:=20; c:=2; R<x>:=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
%Y A007405 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).
%Y A007405 Cf. A004211, A039755, A039756, A187251.
%K A007405 nonn,nice
%O A007405 0,2
%A A007405 _N. J. A. Sloane_
%E A007405 Name edited by _G. C. Greubel_, Feb 24 2019