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Convolution of A115140 with itself.
14

%I #43 Sep 08 2022 08:45:23

%S 1,-2,-1,-2,-5,-14,-42,-132,-429,-1430,-4862,-16796,-58786,-208012,

%T -742900,-2674440,-9694845,-35357670,-129644790,-477638700,

%U -1767263190,-6564120420,-24466267020,-91482563640,-343059613650,-1289904147324,-4861946401452,-18367353072152

%N Convolution of A115140 with itself.

%C 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]

%C a(n)*(-1)^n is the A-sequence for the Riordan triangle A111125. - _Wolfdieter Lang_, Jun 26 2011

%H Vincenzo Librandi, <a href="/A115141/b115141.txt">Table of n, a(n) for n = 0..1000</a>

%F 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).

%F 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.

%F The convolution inverse is A000108(x)^2. - _Michael Somos_, Mar 28 2012

%F REVERT transform is A069271. - _Michael Somos_, Mar 28 2012

%F EULER transform of -A060165. - _Michael Somos_, Mar 28 2012

%F D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1)=0. - _R. J. Mathar_, Feb 21 2020

%e 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 + ...

%t 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 *)

%t CoefficientList[Series[(1-2*x+Sqrt[1-4*x])/2, {x, 0, 30}], x] (* _G. C. Greubel_, Feb 12 2019 *)

%o (PARI) {a(n) = if( n<1, n==0, -(n==1) -binomial( 2*n-2, n-1) / n)} /* _Michael Somos_, Mar 28 2012 */

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-2*x+Sqrt(1-4*x))/2 )); // _G. C. Greubel_, Feb 12 2019

%o (Sage) ((1-2*x+sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 12 2019

%Y Cf. A000108, A115140, A115142 - A115149.

%K sign,easy

%O 0,2

%A _Wolfdieter Lang_, Jan 13 2006