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 A007405 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=2. (Formerly M1674) 22

%I M1674

%S 1,2,6,24,116,648,4088,28640,219920,1832224,16430176,157554048,

%T 1606879040,17350255744,197553645440,2363935624704,29638547505408,

%U 388328781668864,5304452565517824,75381218537805824,1112348880749130752,17014743624340539392,269360902955086379008

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

%C Binomial transform of A004211.

%C 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

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

%H T. D. Noe, <a href="/A007405/b007405.txt">Table of n, a(n) for n = 0..100</a>

%H V. E. Adler, <a href="http://arxiv.org/abs/1510.02900">Set partitions and integrable hierarchies</a>, arXiv:1510.02900 [nlin.SI], 2015.

%H P. 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 P. 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 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 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 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 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 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F E.g.f.: exp(x + (exp(2*x) - 1)/2).

%F Row sums of triangles A039755, A039756. - _Philippe Deléham_, Feb 20 2005

%F 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 1, 1, 0, 0, 0, 0, ...

%F 2, 1, 1, 0, 0, 0, ...

%F 4, 4, 1, 1, 0, 0, ...

%F 8, 12, 6, 1, 1, 0, ...

%F 16, 32, 24, 8, 1, 1, ...

%F ... - _Gary W. Adamson_, Aug 01 2011

%F 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 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 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 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 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 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 a(n) = exp(-1/2) * Sum_{k>=0} (2*k + 1)^n / (2^k * k!). - _Ilya Gutkovskiy_, Apr 16 2020

%F a(n) = Sum_{k=0..n} binomial(n,k) * A187251(k) * A187251(n-k). - _Vaclav Kotesovec_, Apr 17 2020

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

%p a:=series(exp(x+(exp(2*x)-1)/2),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # _Paolo P. Lava_, May 23 2019

%t 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 Table[Sum[Binomial[n, k] * 2^k * BellB[k, 1/2], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 17 2020 *)

%o (PARI) x='x+O('x^66); Vec(serlaplace(exp(x+1/2*exp(2*x)-1/2))) \\ _Joerg Arndt_, May 13 2013

%o (Sage)

%o @CachedFunction

%o def S(n, k, m):

%o if k > n or k < 0 : return 0

%o if n == 0 and k == 0: return 1

%o return S(n-1, k-1, m) + (m*(k+1)-1)*S(n-1, k, m)

%o def A007405(n): return add(S(n, k, 2) for k in (0..n)) # _Peter Luschny_, May 20 2013

%o (Sage)

%o b=2;

%o def A007405_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

%o return P( exp(x +(exp(b*x)-1)/b) ).egf_to_ogf().list()

%o A007405_list(30) # _G. C. Greubel_, Feb 24 2019

%o (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 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).

%K nonn,nice

%O 0,2

%A _N. J. A. Sloane_

%E Name edited by _G. C. Greubel_, Feb 24 2019

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Last modified April 15 20:40 EDT 2021. Contains 342977 sequences. (Running on oeis4.)