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A115141 Convolution of A115140 with itself. 11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is the so-called A-sequence for the Riordan triangles A053122, A158454, A129818 and A110162. 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, and Nov 22 2012]

a(n)*(-1)^n is the A-sequence for the Riordan triangle A111125. [Wolfdieter Lang, Jun 26 2011]

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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.

The convolution inverse is A000108(x)^2. - Michael Somos, Mar 28 2012

REVERT transform is A069271. - Michael Somos, Mar 28 2012

EULER transform of -A060165. - Michael Somos, Mar 28 2012

EXAMPLE

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-Fran├žois Alcover, Oct 21 2011 *)

PROG

(PARI) {a(n) = if( n<1, n==0, -(n==1) -binomial( 2*n-2, n-1) / n)} /* Michael Somos, Mar 28 2012 */

CROSSREFS

Cf. A000108.

Sequence in context: A107087 A279955 A280339 * A031148 A032238 A000619

Adjacent sequences:  A115138 A115139 A115140 * A115142 A115143 A115144

KEYWORD

sign,easy

AUTHOR

Wolfdieter Lang, Jan 13 2006

STATUS

approved

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Last modified February 25 18:12 EST 2018. Contains 299655 sequences. (Running on oeis4.)