A115140 O.g.f. inverse of Catalan A000108 o.g.f.

1, -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

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1668

Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.

Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.

Formula

O.g.f.: 1/c(x) = 1-x*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers).

a(0) = 1, a(n) = -C(n-1), n>=1, with C(n):=A000108(n) (Catalan).

G.f.: (1 + sqrt(1-4*x))/2=U(0) where U(k)=1 - x/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 29 2012

G.f.: 1/G(0) where G(k) = 1 - x/(x - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 12 2012

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

D-finite with recurrence n*a(n) +2*(-2*n+3)*a(n-1)=0. a(n) = A002420(n)/2, n>0. - R. J. Mathar, Aug 09 2015

a(n) ~ -2^(2*n-2) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, May 06 2021

Crossrefs

See A034807 and A115149 for comments.

For convolutions of this sequence see A115141-A115149.

Sequence in context: A287973 A291825 A287974 * A120588 A168491 A000108

Adjacent sequences: A115137 A115138 A115139 * A115141 A115142 A115143

Keyword

sign,easy

Author

Wolfdieter Lang, Jan 13 2006

Status

approved