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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. 4
1, 1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753 (list; graph; refs; listen; history; text; internal format)
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).

Recurrence: y(n+2) = 2( 2 n + 1 ) y(n+1) + y(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

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Last modified March 24 00:12 EDT 2019. Contains 321444 sequences. (Running on oeis4.)