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A115143 a(n) = -4*binomial(2*n-5, n-4)/n for n > 0 and a(0) = 1.

%I A115143 #32 Sep 15 2024 14:53:53
%S A115143 1,-4,2,0,-1,-4,-14,-48,-165,-572,-2002,-7072,-25194,-90440,-326876,
%T A115143 -1188640,-4345965,-15967980,-58929450,-218349120,-811985790,
%U A115143 -3029594040,-11338026180,-42550029600,-160094486370,-603784920024,-2282138106804,-8643460269248,-32798844771700
%N A115143 a(n) = -4*binomial(2*n-5, n-4)/n for n > 0 and a(0) = 1.
%C A115143 Previous name: Fourth convolution of A115140.
%C A115143 a(n+4) := - convolution ( A000108(n+1) ), n=0,1,... - _Tilman Neumann_, Jan 05 2009
%C A115143 Self-convolution of A115141. - _R. J. Mathar_, Sep 26 2012
%H A115143 Seiichi Manyama, <a href="/A115143/b115143.txt">Table of n, a(n) for n = 0..1669</a>
%F A115143 O.g.f.: 1/c(x)^4 = P(5, x) - x*P(4, x)*c(x) with the o.g.f. c(x) := (1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(5, x) = 1-3*x+x^2 and P(4, x) = 1-2*x.
%F A115143 a(n) = -C4(n-4), n>=4, with C4(n) := A002057(n) (fourth convolution of Catalan numbers). a(0)=1, a(1)=-4, a(2)=2, a(3)=0. [1, -4, 2] is row n=4 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
%F A115143 E.g.f.: 1 - 3*x + 1/2*x^2 - x*Q(0), where Q(k)= 1 - 2*x/(k+2 - (k+2)*(2*k+1)/(2*k+1 - (k+2)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 28 2013
%F A115143 D-finite with recurrence n*(n-4)*a(n) -2*(2*n-5)*(n-3)*a(n-1)=0. - _R. J. Mathar_, Sep 15 2024
%p A115143 A115143 := n -> `if`(n=0, 1, -4*binomial(2*n-5,n-4)/n):
%p A115143 seq(A115143(n), n=0..28); # _Peter Luschny_, Feb 27 2017
%p A115143 A115143List := proc(m) local A, P, n; A := [1,-4,2,0]; P := [-1,0];
%p A115143 for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
%p A115143 A := [op(A), P[-1]] od; A end: A115143List(27); # _Peter Luschny_, Mar 26 2022
%t A115143 Join[{1},Table[-4*Binomial[2n-5,n-4]/n,{n,30}]] (* _Harvey P. Dale_, Dec 01 2017 *)
%t A115143 CoefficientList[Series[(1-4*x+2*x^2+(1-2*x)*Sqrt[1-4*x])/2, {x,0,30}], x] (* _G. C. Greubel_, Feb 12 2019 *)
%o A115143 (PARI) my(x='x+O('x^30)); Vec((1-4*x+2*x^2 +(1-2*x)*sqrt(1-4*x))/2) \\ _G. C. Greubel_, Feb 12 2019
%o A115143 (Magma) [1,-4,2] cat [-4*Binomial(2*n-5,n-4)/n: n in [3..30]]; // _G. C. Greubel_, Feb 12 2019
%o A115143 (Sage) [1,-4,2] + [-4*binomial(2*n-5,n-4)/n for n in (3..30)] # _G. C. Greubel_, Feb 12 2019
%Y A115143 Cf. A115139 - A115142, A115144 - A115149, A099376, A000108.
%K A115143 sign,easy
%O A115143 0,2
%A A115143 _Wolfdieter Lang_, Jan 13 2006
%E A115143 Simpler name from _Peter Luschny_, Feb 27 2017