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a(n) = (1/2)*s(n+3), where s = A025251.
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%I #14 Dec 27 2023 12:36:28

%S 0,1,2,1,6,9,12,41,60,121,310,505,1162,2577,4760,11089,23256,47089,

%T 107274,223345,476366,1061017,2237796,4888313,10745748,23048169,

%U 50792638,111180265,241786898,534219297,1170798128,2570337441,5684509232,12503504353,27613172114

%N a(n) = (1/2)*s(n+3), where s = A025251.

%F G.f.: (1 - x^2 - 4*x^3 - sqrt(1 - 2*x^2 - 8*x^3 + x^4)) / (4*x^3). - _Michael Somos_, Jun 08 2000

%F (n+3)*a(n) +(n+2)*a(n-1) -2*n*a(n-2) +2*(-5*n+7)*a(n-3) +(-7*n+17)*a(n-4) +(n-4)*a(n-5)=0. - _R. J. Mathar_, Dec 15 2013

%F 0 = a(n)*(+a(n+1) - 20*a(n+2) - 8*a(n+3) + 7*a(n+5)) +a(n+1)*(+4*a(n+1) + 68*a(n+2) + 40*a(n+3) - 5*a(n+4) - 44*a(n+5)) + a(n+2)*(-8*a(n+2) + 4*a(n+3) + 28*a(n+4) - 8*a(n+5)) + a(n+3)*(+4*a(n+4)) + a(n+4)*(+a(n+5)) for all n>0. - _Michael Somos_, Feb 08 2015

%F a(n) = A160565(n) for all n>0. - _Michael Somos_, Feb 08 2015

%e G.f. = x^2 + 2*x^3 + x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 41*x^8 + 60*x^9 + ...

%t a[ n_] := SeriesCoefficient[ (1 - x^2 - 4 x^3 - Sqrt[1 - 2 x^2 - 8 x^3 + x^4]) / (4 x^3), {x, 0, n}]; (* _Michael Somos_, Feb 08 2015 *)

%o (PARI) {a(n) = if( n<1, 0, polcoeff( (-sqrt(1 - 2*x^2 - 8*x^3 + x^4 + x^4*O(x^n))) / 4, n+3))};

%Y Cf. A025251, A160565.

%K nonn

%O 1,3

%A _Clark Kimberling_