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

0,3

Links

Harvey P. Dale, Table of n, a(n) for n = 0..380

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties , arXiv:1103.2582 [math.CO], 2011-2013.

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

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 *)

Crossrefs

Cf. A001006.

Sequence in context: A059280 A085295 A094956 * A143768 A256493 A353256

Adjacent sequences: A080891 A080892 A080893 * A080895 A080896 A080897

Keyword

easy,nice,nonn

Author

Emanuele Munarini, Mar 31 2003

Status

approved