login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007405 Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=2.
(Formerly M1674)
22
1, 2, 6, 24, 116, 648, 4088, 28640, 219920, 1832224, 16430176, 157554048, 1606879040, 17350255744, 197553645440, 2363935624704, 29638547505408, 388328781668864, 5304452565517824, 75381218537805824, 1112348880749130752, 17014743624340539392, 269360902955086379008 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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.

P. 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.

P. 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.

Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

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

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

EXAMPLE

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

MAPLE

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

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 *)

PROG

(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.<x> = 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

(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

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).

Sequence in context: A210591 A342141 A266332 * A324130 A324131 A221988

Adjacent sequences:  A007402 A007403 A007404 * A007406 A007407 A007408

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 2 12:26 EST 2021. Contains 341750 sequences. (Running on oeis4.)