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
This is the number of type B set partitions, see R. Suter. - Per W. Alexandersson, Dec 19 2022
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.
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.
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.
FORMULA
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, ...
... - 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) ~ 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
EXAMPLE
a(4) = 116 = sum of top row terms of M^3 = (49 + 44 + 18 + 4 + 1).
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
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Name edited by G. C. Greubel, Feb 24 2019
STATUS
approved