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

0,4

Links

Table of n, a(n) for n=0..18.

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 - 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)

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 *)

Prog

(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 */

Crossrefs

Cf. A005043.

Sequence in context: A349117 A125796 A135745 * A047899 A229041 A263247

Adjacent sequences: A080892 A080893 A080894 * A080896 A080897 A080898

Keyword

easy,nice,nonn

Author

Emanuele Munarini, Mar 31 2003

Status

approved