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A025273 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5, starting 1,0,1,1.

%I A025273 #19 Jan 13 2025 11:13:22
%S A025273 1,0,1,1,2,5,12,29,72,182,466,1207,3158,8334,22158,59299,159614,
%T A025273 431838,1173710,3203244,8774780,24118522,66497316,183858411,509670494,
%U A025273 1416231616,3944027402,11006186760,30772507128,86191006746,241815195292,679488418879
%N A025273 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5, starting 1,0,1,1.
%C A025273 The binomial transform of A025250(n+1) is A025273(n+2). - _Paul Barry_, May 11 2005
%H A025273 Robert Israel, <a href="/A025273/b025273.txt">Table of n, a(n) for n = 1..2140</a>
%H A025273 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 A025273 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 A025273 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 A025273 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 A025273 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 A025273 map(f, [$0..50]); # _Robert Israel_, Nov 02 2016
%p A025273 # alternative
%p A025273 A025273 := proc(n)
%p A025273     option remember ;
%p A025273     if n < 5 then
%p A025273         op(n,[1,0,1,1]) ;
%p A025273     else
%p A025273         add( procname(i)*procname(n-i),i=1..n-1) ;
%p A025273     end if;
%p A025273 end proc:
%p A025273 seq(A025273(n),n=1..20) ; # _R. J. Mathar_, Jan 13 2025
%t A025273 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 A025273 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 A025273 nonn
%O A025273 1,5
%A A025273 _Clark Kimberling_