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a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5, starting 1,1,1,0.
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%I #19 Jan 13 2025 11:14:53

%S 1,1,1,0,2,5,14,42,122,360,1068,3181,9526,28654,86558,262528,799212,

%T 2441538,7483052,23004500,70921492,219226064,679328952,2109948221,

%U 6567539814,20483936790,64010196918,200382350016,628344541644,1973428795542

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

%F G.f.: (1-sqrt(1-4x+4x^3+12x^4))/2.

%F Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 2*(2*n-9)*a(n-3) - 12*(n-6)*a(n-4). - _Vaclav Kotesovec_, Jan 25 2015

%p A025274 := proc(n)

%p option remember ;

%p if n < 5 then

%p op(n,[1,1,1,0]) ;

%p else

%p add( procname(i)*procname(n-i),i=1..n-1) ;

%p end if;

%p end proc:

%p seq(A025274(n),n=1..20) ; # _R. J. Mathar_, Jan 13 2025

%t nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = 1; aa[[3]] = 1; aa[[4]] = 0; Do[aa[[n]] = Sum[aa[[k]]*aa[[n-k]],{k,1,n-1}],{n,5,nmax}]; aa (* _Vaclav Kotesovec_, Jan 25 2015 *)

%t Rest[CoefficientList[Series[(1-Sqrt[1-4x+4x^3+12x^4])/2,{x,0,30}],x]] (* _Harvey P. Dale_, Aug 21 2024 *)

%o (PARI) default(seriesprecision, 100); Vec((1-sqrt(1-4*x+4*x^3+12*x^4))/2 + O(x^50)) \\ _Michel Marcus_, Nov 22 2014

%K nonn,eigen

%O 1,5

%A _Clark Kimberling_