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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. 1
 1, 1, 3, 19, 169, 2001, 29371, 516643, 10590609, 248113729, 6541248691, 191719042131, 6185020391353, 217824649952209, 8316522297035499, 342188317852814371, 15095509523107176481, 710794856254145560833 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 13 00:40 EDT 2024. Contains 375857 sequences. (Running on oeis4.)