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%I #9 Jun 07 2014 10:35:55
%S 1,1,3,20,215,3019,50750,978792,21118641,501447228,12954812178,
%T 361153740034,10796374169997,344373435613191,11672904804892441,
%U 419017404782400368,15881588444144243375,633896023527831462147,26581090260348132921162,1168473922210084561137995
%N G.f. satisfies: a(n) = [x^n] ( A(x)^(n+1) + 1/A(x)^(n+1) ) / (n+1) for n>1, with a(0)=a(1)=1.
%F G.f. satisfies: A(x) = x-1 + Series_Reversion(x*A(x))/x + Series_Reversion(x/A(x))/x.
%e G.f.: A(x) = 1 + x + 3*x^2 + 20*x^3 + 215*x^4 + 3019*x^5 + 50750*x^6 +...
%e Form a table of coefficients in A(x)^n + 1/A(x)^n for n>=1, which begins:
%e [2, 0, 1, 5, 41, 474, 6816, 115062, 2209419, ...];
%e [2, 0, 4, 20, 165, 1906, 27371, 461606, 8857729, ...];
%e [2, 0, 9, 45, 375, 4326, 61987, 1043721, 20005287, ...];
%e [2, 0, 16, 80, 676, 7784, 111204, 1868272, 35753349, ...];
%e [2, 0, 25, 125, 1075, 12350, 175785, 2944975, 56245065, ...];
%e [2, 0, 36, 180, 1581, 18114, 256723, 4286502, 81666879, ...];
%e [2, 0, 49, 245, 2205, 25186, 355250, 5908616, 112250341, ...];
%e [2, 0, 64, 320, 2960, 33696, 472848, 7830336, 148274340, ...];
%e [2, 0, 81, 405, 3861, 43794, 611262, 10074132, 190067769, ...]; ...
%e then this sequence is obtained from the main diagonal after the initial 2 terms:
%e [2, 0, 9/3, 80/4, 1075/5, 18114/6, 355250/7, 7830336/8, 190067769/9, ...].
%e RELATED SERIES.
%e Define F(x) = A(x/F(x)):
%e F(x) = 1 + x + 2*x^2 + 13*x^3 + 142*x^4 + 1988*x^5 + 33250*x^6 +...
%e 1/F(x) = 1 - x - x^2 - 10*x^3 - 117*x^4 - 1696*x^5 - 29060*x^6 -...
%e and G(x) = A(x*G(x)):
%e G(x) = 1 + x + 4*x^2 + 30*x^3 + 332*x^4 + 4715*x^5 + 79810*x^6 +...
%e then A(x) = x-1 + G(x) + 1/F(x).
%o (PARI) {a(n)=local(A=vector(n+1, i, 1)); for(n=2, #A-1, A[n+1] = Vec(Ser(A)^(n+1) + 1/Ser(A)^(n+1))[n+1]/(n+1)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* Build vector of n=0..30 terms (faster): */
%o {A=vector(31, i, 1); for(n=2, #A-1, A[n+1] = Vec(Ser(A)^(n+1) + 1/Ser(A)^(n+1))[n+1]/(n+1)); A}
%Y Cf. A237644.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 04 2014
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