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%I #17 Oct 17 2018 15:43:45
%S 1,0,1,1,2,5,12,29,72,182,466,1207,3158,8334,22158,59299,159614,
%T 431838,1173710,3203244,8774780,24118522,66497316,183858411,509670494,
%U 1416231616,3944027402,11006186760,30772507128,86191006746,241815195292,679488418879
%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5.
%C The binomial transform of A025250(n+1) is A025273(n+2). - _Paul Barry_, May 11 2005
%H Robert Israel, <a href="/A025273/b025273.txt">Table of n, a(n) for n = 1..2140</a>
%H Paul Barry, <a href="https://arxiv.org/abs/1807.05794">Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences</a>, arXiv:1807.05794 [math.CO], 2018.
%F G.f. (with offset 0 instead of 1): (1-sqrt(1-4*x+4*x^2-4*x^3+4*x^4))/(2*x). - _Paul Barry_, May 11 2005
%F Conjecture: (with offset 0 instead of 1) (n+1)*a(n) +2*(-2*n+1)*a(n-1) +4*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +4*(n-5)*a(n-4)=0. - _R. J. Mathar_, Nov 24 2012
%F Conjecture follows from the differential equation 4*x^3-3*x^2+2*x-1+(-4*x^4+2*x^3-2*x+1)*g(x)+(4*x^5-4*x^4+4*x^3-4*x^2+x)*g'(x)=0 satisfied by the g.f. - _Robert Israel_, Nov 02 2016
%p f:= gfun:-rectoproc({(n+1)*a(n) +2*(-2*n+1)*a(n-1) +4*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +4*(n-5)*a(n-4)=0, a(0)=1,a(1)=0,a(2)=1,a(3)=1},a(n),remember):
%p map(f, [$0..50]); # _Robert Israel_, Nov 02 2016
%t nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = 0; aa[[3]] = 1; aa[[4]] = 1; Do[aa[[n]] = Sum[aa[[k]]*aa[[n-k]],{k,1,n-1}],{n,5,nmax}]; aa (* _Vaclav Kotesovec_, Jan 25 2015 *)
%t CoefficientList[Series[(1-Sqrt[1-4*x+4*x^2-4*x^3+4*x^4])/(2*x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 01 2015 *)
%K nonn
%O 1,5
%A _Clark Kimberling_
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