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.

1, 1, 3, 19, 169, 2001, 29371, 516643, 10590609, 248113729, 6541248691, 191719042131, 6185020391353, 217824649952209, 8316522297035499, 342188317852814371, 15095509523107176481, 710794856254145560833

Offset: 0

Author

Emanuele Munarini, Mar 31 2003

Keywords

• easy
• nice
• nonn

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..380
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A001006.

Programs

Mathematica

#/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 *)

Formula

E.g.f.: exp((1 - x - sqrt(1 - 2*x - 3*x^2))/(2x)).

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

a(n) ~ 3^(n+1/2)*n^(n-1)/(sqrt(2)*exp(n-1)). - Vaclav Kotesovec, Oct 05 2013

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