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A007405
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Dowling numbers: e.g.f.: exp(x + (exp(b*x) - 1)/b) with b=2.
(Formerly M1674)
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26
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1, 2, 6, 24, 116, 648, 4088, 28640, 219920, 1832224, 16430176, 157554048, 1606879040, 17350255744, 197553645440, 2363935624704, 29638547505408, 388328781668864, 5304452565517824, 75381218537805824, 1112348880749130752, 17014743624340539392, 269360902955086379008
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
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FORMULA
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E.g.f.: exp(x + (exp(2*x) - 1)/2).
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, ...
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) ~ 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
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EXAMPLE
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a(4) = 116 = sum of top row terms of M^3 = (49 + 44 + 18 + 4 + 1).
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MATHEMATICA
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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 *)
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PROG
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(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)
(Sage)
b=2;
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(x +(exp(b*x)-1)/b) ).egf_to_ogf().list()
(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
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CROSSREFS
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KEYWORD
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nonn,nice,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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