VOOZH about

URL: https://oeis.org/A115141

⇱ A115141 - OEIS

login
A115141
Convolution of A115140 with itself.
15
1, -2, -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
0,2
COMMENTS
This is the so-called A-sequence for the Riordan triangles A053122, A110162, A129818, A158454 and signed A158909. For the notion of Z- and A-sequences for Riordan arrays see the W. Lang link under A006232 with details and references. - Wolfdieter Lang, Dec 20 2010. [Revised, Nov 13 2012, Nov 22 2012 and Oct 22 2019]
a(n)*(-1)^n is the A-sequence for the Riordan triangle A111125. - Wolfdieter Lang, Jun 26 2011
LINKS
FORMULA
O.g.f.: 1/c(x)^2 = (1-x) - 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(1) = -2, a(n) = -C(n-1), n>=2, with C(n) = A000108(n) (Catalan). The start [1, -2] is row n=2 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
From Michael Somos, Mar 28 2012: (Start)
The convolution inverse is A000108(x)^2.
REVERT transform is A069271.
EULER transform of -A060165. (End)
D-finite with recurrence: n*a(n) + 2*(-2*n+3)*a(n-1) = 0. - R. J. Mathar, Feb 21 2020
a(n) ~ -4^(n-1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 05 2025
EXAMPLE
G.f. = 1 - 2*x - x^2 - 2*x^3 - 5*x^4 - 14*x^5 - 42*x^6 - 132*x^7 - 429*x^8 + ...
MATHEMATICA
a[n_] := -First[ ListConvolve[ cc = Array[ CatalanNumber, n-1, 0], cc]]; a[0] = 1; a[1] = -2; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 21 2011 *)
CoefficientList[Series[(1-2*x+Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)
PROG
(PARI) {a(n) = if( n<1, n==0, -(n==1) -binomial( 2*n-2, n-1) / n)} /* Michael Somos, Mar 28 2012 */
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-2*x+Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
(SageMath) ((1-2*x+sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, Jan 13 2006
STATUS
approved