A080893 E.g.f. exp(x*C(x)) = exp((1-sqrt(1-4*x))/2), where C(x) is the g.f. of the Catalan numbers A000108.

1, 1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753

Offset

0,3

Comments

Essentially the same as A001517: a(n+1) = A001517(n).

Links

Table of n, a(n) for n=0..17.

W. Mlotkowski, A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.

Formula

E.g.f.: exp((1-sqrt(1-4*x))/2).

D-finite with recurrence: a(n+2) = 2( 2 n + 1 ) a(n+1) + a(n)

Recurrence: y(n+1) = sum(k=0..n, binomial(n, k)*binomial(2k, k)*k!*y(n-k) ).

a(1 - n) = a(n). a(n + 1) = A001517(n). - Michael Somos, Apr 07 2012

G.f.: 1 + x/Q(0), where Q(k)= 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013

a(n) ~ 2^(2*n-3/2)*n^(n-1)/exp(n-1/2). - Vaclav Kotesovec, Jun 26 2013

a(n) = hypergeom([-n+1, n], [], -1). - Peter Luschny, Oct 17 2014

Mathematica

y[x_] := y[x] = 2(2x - 3)y[x - 1] + y[x - 2]; y[0] = 1; y[1] = 1; Table[y[n], {n, 0, 17}]
With[{nn=20}, CoefficientList[Series[Exp[(1-Sqrt[1-4x])/2], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 30 2011 *)

Prog

(PARI) {a(n) = if( n<1, n = 1 - n); n! * polcoeff( exp( (1 - sqrt(1 - 4*x + x * O(x^n))) / 2), n)} /* Michael Somos, Apr 07 2012 */
(Sage)
A080893 = lambda n: hypergeometric([-n+1, n], [], -1)
[simplify(A080893(n)) for n in (0..19)] # Peter Luschny, Oct 17 2014

Crossrefs

Cf. A000108, A001517.

Sequence in context: A155805 A218261 A001517 * A028854 A222865 A108292

Adjacent sequences: A080890 A080891 A080892 * A080894 A080895 A080896

Keyword

easy,nice,nonn

Author

Emanuele Munarini, Mar 31 2003

Status

approved