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%I #11 Jun 17 2020 07:33:41
%S 1,1,2,6,18,55,171,538,1708,5461,17560,56728,183973,598597,1953145,
%T 6388376,20939664,68764283,226192964,745146462,2458020664,8118111977,
%U 26841209903,88835163150,294284206183,975699571009,3237456793478,10749922312752,35718863630895,118757413662397
%N Alternating row sums of triangle A291844.
%H Gheorghe Coserea, <a href="/A294159/b294159.txt">Table of n, a(n) for n = 0..303</a>
%F G.f. y(x) satisfies: 0 = (x^3 + x^2 + 3*x - 1)*(y^2 - y) + x.
%F Conjecture: D-finite with recurrence n*a(n) +(-3*n+1)*a(n-1) +2*(-n+3)*a(n-2) +2*(n-5)*a(n-3) +(n-4)*a(n-4) +(n-5)*a(n-5)=0. - _R. J. Mathar_, Jun 17 2020
%o (PARI)
%o A291843_ser(N, t='t) = {
%o my(x='x+O('x^N), y=1, y1=0, n=1,
%o dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
%o while (n++,
%o y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
%o (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
%o if (y1 == y, break); y = y1; ); y;
%o };
%o A291844_ser(N, t='t) = {
%o my(z = A291843_ser(N+1, t));
%o ((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
%o };
%o Vec(A291844_ser(30,-1))
%o \\ test: y=A291844_ser(200,-1); 0==(x^3 + x^2 + 3*x - 1)*(y^2 - y) + x
%Y Cf. A291844.
%K nonn
%O 0,3
%A _Gheorghe Coserea_, Nov 03 2017
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