A025168 - OEIS

%I #79 Aug 05 2024 22:00:46

%S 1,1,5,37,361,4361,62701,1044205,19748177,417787921,9770678101,

%T 250194150581,6959638411705,208919770666777,6729933476435261,

%U 231512615111396221,8469125401589550241,328241040596380393505,13434223364220816489637,578931271898150002093381

%N Expansion of e.g.f.: exp(x/(1-2*x)).

%C From _Peter Bala_, Nov 21 2017: (Start)

%C The sequence terms have the form 4*m + 1 (follows from the recurrence).

%C For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k (proof by induction on n making use of the recurrence - the starting case a(k) == a(0) (mod k) for all k follows from the sum formula for a(k)). Hence for each k, the sequence b(n) == a(n) (mod k) is periodic with the exact period dividing k. (End)

%C Compound Poisson distribution with parameter 1 and distribution Geometric(1/2) has a probability mass function p_n = a(n)*e^(-1/2)/(4^n*n!). More specifically, let S = Sum_{i=0..N} X_i where X_i's are i.i.d. random variables with Geometric(1/2) distribution (i.e., Pr{X_i = k} = 1/2^(k+1) for k=0,1,2...) and N is a random variable with Poisson(1) distribution independent of all X_i's. Then Pr{S=n} = a(n)*e^(-1/2)/(4^n*n!) = a(n)*e^(-1/2)/A047053(n) for nonnegative integers n. - _Xiaohan Zhang_, Nov 16 2022

%H Robert Israel, <a href="/A025168/b025168.txt">Table of n, a(n) for n = 0..400</a>

%H Norihiro Nakashima and Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.

%H K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a>, arXiv:quant-ph/0312202, 2003; J. Phys. A 37 (2004), 3475-3487.

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

%H N. J. A. Sloane and Thomas Wieder, <a href="https://arxiv.org/abs/math/0307064">The Number of Hierarchical Orderings</a>, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

%H Thomas Wieder, <a href="/A103446/a103446.txt">Expanded definitions of A103446 and A025168</a>

%F Second LAH transform of A000012. LAH transform of A000262. a(n) = Sum_{k=0..n} 2^(n-k)*n!/k!*binomial(n-1, k-1). - _Vladeta Jovovic_, Oct 17 2003

%F Define f_1(x), f_2(x), ... such that f_1(x) = e^x, f_{n+1}(x) = (d/dx)(x^2*f_n(x)), for n=2,3,.... Then a(n) = e^(-1/2)*4*(n-1)*f_n(1/2). - _Milan Janjic_, May 30 2008

%F From _Vaclav Kotesovec_, Jun 22 2013: (Start)

%F D-finite with recurrence: a(n) = (4*n-3)*a(n-1) - 4*(n-2)*(n-1)*a(n-2).

%F a(n) ~ 2^(n-3/4)*n^(n-1/4)*exp(sqrt(2*n)-n-1/4) * (1-1/(3*sqrt(2*n))).

%F (End)

%F E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - x/(x + (k+1)*(1-2*x)/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 09 2013

%F a(n) = hypergeometric([-n,-n+1],[],2). - _Peter Luschny_, Sep 22 2014

%F Sum_{n>=0} a(n)/(4^n*n!) = sqrt(e) = A019774. -_Xiaohan Zhang_, Nov 16 2022

%p with(combstruct); SetSeqSeqL := [T, {T=Set(S), S=Sequence(U,card >= 1), U=Sequence(Z,card >=1)},labeled];

%p f:= gfun:-rectoproc({a(n) = (4*n-3)*a(n-1) - 4*(n-2)*(n-1)*a(n-2),a(0)=1,a(1)=1},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, Nov 21 2017

%t Table[ n! 2^n LaguerreL[ n, 1, -1/2 ], {n, 0, 12} ]

%t With[{nn=20},CoefficientList[Series[Exp[x/(1-2x)],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 12 2012 *)

%o (Sage)

%o A025168 = lambda n: hypergeometric([-n,-n+1], [], 2)

%o [Integer(A025168(n).n(100)) for n in range(20)] # _Peter Luschny_, Sep 22 2014

%Y Cf. A000012, A000262, A103446.

%K nonn,easy

%O 0,3

%A _Wouter Meeussen_

%E Corrected and extended by _Vladeta Jovovic_, Sep 08 2002