cp's OEIS Frontend

A080895 Expansion of the exponential series exp( x R(x) ) = exp((1 + x - sqrt(1 - 2 x - 3x^2))/(2(1 + x))), where R(x) is the ordinary generating series of the Riordan numbers A005043.

%I A080895 #27 Oct 08 2017 17:32:12
%S A080895 1,1,1,7,49,541,7321,122011,2390977,54027289,1382140081,39493358191,
%T A080895 1246693438321,43087256236597,1618203187947529,65621724413560771,
%U A080895 2857736621103221761,133014764141210620081,6589916027200886776417
%N A080895 Expansion of the exponential series exp( x R(x) ) = exp((1 + x - sqrt(1 - 2 x - 3x^2))/(2(1 + x))), where R(x) is the ordinary generating series of the Riordan numbers A005043.
%H A080895 Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A080895 E.g.f.: exp((1 + x - sqrt(1 - 2 x - 3x^2))/(2(1 + x))).
%F A080895 a(n) = (n-1)!*Sum_{k=1..n} ((Sum_{j=k..n} C(n,j)*C(2*j-k-1, j-1)*(-1)^(n-j))/(k-1)!), n > 0. - _Vladimir Kruchinin_, Sep 07 2010
%F A080895 a(n) ~ sqrt(2)*3^(n + 1/2)*n^(n-1)/(8*exp(n - 1/2)). - _Vaclav Kotesovec_, Sep 29 2013
%F A080895 From _Benedict W. J. Irwin_, May 27 2016: (Start)
%F A080895 Let y(0)=1, y(1)=1, y(2)=1/2, y(3)=7/6,
%F A080895 Let -3n*(1+n)*y(n) - (12+20n+8n^2)*y(n+1) - (25+24n+6n^2)*y(n+2)+(n+3)*(n+4)*y(n+4) = 0,
%F A080895 a(n) = n!*y(n).
%F A080895 (End)
%t A080895 a[n_] := (n-1)!*Sum[ ((-1)^(n+k)*Binomial[n, k]* HypergeometricPFQ[ {k/2 + 1/2, k/2, k-n}, {k, k+1}, 4])/(k-1)!, {k, 1, n}]; a[0] = 1; Table[ a[n], {n, 0, 18}] (* _Jean-François Alcover_, Dec 20 2011, after _Vladimir Kruchinin_ *)
%o A080895 (Maxima) a(n):=(n-1)!*sum(sum(binomial(n,j)*binomial(2*j-k-1,j-1)*(-1)^(n-j), j,k,n)/(k-1)!, k,1,n); /* _Vladimir Kruchinin_, Sep 07 2010 */
%Y A080895 Cf. A005043.
%K A080895 easy,nice,nonn
%O A080895 0,4
%A A080895 _Emanuele Munarini_, Mar 31 2003