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.

1, 1, 1, 7, 49, 541, 7321, 122011, 2390977, 54027289, 1382140081, 39493358191, 1246693438321, 43087256236597, 1618203187947529, 65621724413560771, 2857736621103221761, 133014764141210620081, 6589916027200886776417

Offset: 0

Author

Emanuele Munarini, Mar 31 2003

Keywords

• easy
• nice
• nonn

Links

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

Crossrefs

Cf. A005043.

Programs

Mathematica

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 *)

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 */

Formula

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

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

a(n) ~ sqrt(2)*3^(n + 1/2)*n^(n-1)/(8*exp(n - 1/2)). - Vaclav Kotesovec, Sep 29 2013

From Benedict W. J. Irwin, May 27 2016: (Start)

Let y(0)=1, y(1)=1, y(2)=1/2, y(3)=7/6,

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,

a(n) = n!*y(n).

(End)