A168491 a(n) = (-1)^n*Catalan(n).

1, -1, 2, -5, 14, -42, 132, -429, 1430, -4862, 16796, -58786, 208012, -742900, 2674440, -9694845, 35357670, -129644790, 477638700, -1767263190, 6564120420, -24466267020, 91482563640, -343059613650, 1289904147324, -4861946401452, 18367353072152, -69533550916004

Offset

0,3

Comments

Second inverse binomial transform of A001405. Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...].

Also the expansion of real root of y+y^2=x, With offset 1, series reversion of x+x^2. - Robert G. Wilson v, Mar 07 2011

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

Index to sequences related to reversion of series

Formula

a(n) = (-1)^n * A000108(n).

G.f.: (sqrt(1+4*x) - 1) / (2*x) = 2 / (sqrt(1+4*x) + 1).

E.g.f.: exp(-2*x)*(BesselI(0, 2*x) + BesselI(1, 2*x)). - Peter Luschny, Aug 26 2012

D-finite with recurrence (n+1)*a(n) +2*(2*n - 1)*a(n-1) = 0. - R. J. Mathar, Oct 06 2012

G.f.: 1 / (1 + x / (1 + x / (1 + x / ...))). - Michael Somos, Jan 03 2013

G.f.: 1/(x*Q(0)) - 1/x, where Q(k)= 1 - (4*k+1)*x/(k+1 - x*(2*k+2)*(4*k+3)/(2*x*(4*k+3) - (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 15 2013

G.f.: G(0)/(2*x) - 1/(2*x), where G(k)= 1 + 4*x*(4*k+1)/( (4*k+2)*(1+4*x) - 2*x*(1+4*x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1+4*x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013

G.f.: G(0)/x - 1/x, where G(k)= k+1 - 2*x*(2*k+1) + 2*x*(k+1)*(2*k+3)/G(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Jul 14 2013

Example

G.f. = 1 - x + 2*x^2 - 5*x^3 + 14*x^4 - 42*x^5 + 132*x^6 - 429*x^7 + ...

Mathematica

CoefficientList[InverseSeries[Series[y + y^2, {y, 0, 28}], x]/x, x] (* Robert G. Wilson v, Mar 07 2011 *)
a[ n_] := If[ n < 0, 0, (-1)^n CatalanNumber[n]]; (* Michael Somos, Nov 22 2014 *)
Table[(-1)^n*CatalanNumber[n], {n, 0, 50] (* G. C. Greubel, Jul 23 2016 *)
Times@@@Partition[Riffle[CatalanNumber[Range[0, 30]], {1, -1}, {2, -1, 2}], 2] (* Harvey P. Dale, Dec 19 2022 *)

Prog

(PARI) a(n)=(-1)^n*binomial(2*n, n)/(n+1); \\ Joerg Arndt, May 15 2013
(Magma) [(-1)^n*Catalan(n): n in [0..40]]; // Vincenzo Librandi, Nov 16 2014

Crossrefs

Sequence in context: A287974 A115140 A120588 * A000108 A057413 A126567

Adjacent sequences: A168488 A168489 A168490 * A168492 A168493 A168494

Keyword

sign,less,easy

Author

Philippe Deléham, Nov 27 2009

Status

approved