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A080894 Expansion of the exponential series exp( x M(x) ) = exp( (1-sqrt(1-2x-3x^2))/(2x) ), where M(x) is the ordinary generating series of the Motzkin numbers A001006.

%I A080894 #28 Feb 19 2025 10:25:21
%S A080894 1,1,3,19,169,2001,29371,516643,10590609,248113729,6541248691,
%T A080894 191719042131,6185020391353,217824649952209,8316522297035499,
%U A080894 342188317852814371,15095509523107176481,710794856254145560833
%N A080894 Expansion of the exponential series exp( x M(x) ) = exp( (1-sqrt(1-2x-3x^2))/(2x) ), where M(x) is the ordinary generating series of the Motzkin numbers A001006.
%H A080894 Harvey P. Dale, <a href="/A080894/b080894.txt">Table of n, a(n) for n = 0..380</a>
%H A080894 Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A080894 E.g.f.: exp((1 - x - sqrt(1 - 2*x - 3*x^2))/(2x)).
%F A080894 a(n) = (n-1)!*Sum_{k=1..n} (1/(k-1)!)*Sum_{j=ceiling((n+k)/2)..n} binomial(n,j)*binomial(j,2*j-n-k). - _Vladimir Kruchinin_, Aug 11 2010
%F A080894 a(n) ~ 3^(n+1/2)*n^(n-1)/(sqrt(2)*exp(n-1)). - _Vaclav Kotesovec_, Oct 05 2013
%F A080894 Conjecture D-finite with recurrence: +(-2*n+3)*a(n) +(-2*n^3+9*n^2-9*n+1)*a(n-1) +(n-1)*(n-2)*(4*n^2-2*n-3)*a(n-2) +3*(n-1)*(n-3)*(2*n-1)*(n-2)^2*a(n-3)=0. - _R. J. Mathar_, Jan 24 2020
%t A080894 #/Sqrt[E]&/@With[{nn=20},CoefficientList[Series[Exp[(1-Sqrt[1-2x-3x^2])/ (2x)],{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Oct 26 2011 *)
%Y A080894 Cf. A001006.
%K A080894 easy,nice,nonn
%O A080894 0,3
%A A080894 _Emanuele Munarini_, Mar 31 2003