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A208975 G.f. satisfies: A(x) = 1 + x*A(x) * [d/dx x/A(x)^2].

%I A208975 #14 Nov 18 2017 05:06:56
%S A208975 1,1,-3,20,-189,2232,-31130,497016,-8907885,176829104,-3849436062,
%T A208975 91187523000,-2335691914050,64344487654800,-1897619527612692,
%U A208975 59667237154623280,-1993022006345620605,70488571028815935072,-2631925423768158446390
%N A208975 G.f. satisfies: A(x) = 1 + x*A(x) * [d/dx x/A(x)^2].
%H A208975 Vaclav Kotesovec, <a href="/A208975/b208975.txt">Table of n, a(n) for n = 0..300</a>
%F A208975 G.f. A(x) satisfies: [x^n] A(x)^(2*n-1) = [x^n] A(x)^(2*n) for n>=2.
%F A208975 G.f.: A(x) = -x/G(-x) where G(x) is the g.f. of A000699, the number of irreducible diagrams with 2n nodes.
%F A208975 a(n) ~ -(-1)^n * 2^(n + 3/2) * n^(n+1) / exp(n+1). - _Vaclav Kotesovec_, Nov 18 2017
%e A208975 G.f.: A(x) = 1 + x - 3*x^2 + 20*x^3 - 189*x^4 + 2232*x^5 - 31130*x^6 +...
%e A208975 Related expansion:
%e A208975 d/dx x/A(x)^2 = 1 - 4*x + 27*x^2 - 248*x^3 + 2830*x^4 - 38232*x^5 +...
%e A208975 Let G(x) be the g.f. of A000699:
%e A208975 G(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 + 38232*x^7 +...
%e A208975 then A(x) = -x/G(-x), or A(x) = 1 + x*A(x) * (x + G(-x))/x^2.
%e A208975 The coefficients in A(x)^n begin:
%e A208975 n=1: [1, 1, -3, 20, -189, 2232,  -31130,  497016,  -8907885, ...];
%e A208975 n=2: [1, 2, -5, 34, -329, 3966,  -56262,  910820, -16509957, ...];
%e A208975 n=3: [1, 3,(-6),43, -429, 5289,  -76350, 1253250, -22971165, ...];
%e A208975 n=4: [1, 4,(-6),48, -497, 6276,  -92214, 1534560, -28436085, ...];
%e A208975 n=5: [1, 5, -5,(50),-540, 6991, -104555, 1763610, -33031335, ...];
%e A208975 n=6: [1, 6, -3,(50),-564, 7488, -113969, 1948038, -36867735, ...];
%e A208975 n=7: [1, 7,  0, 49,(-574),7812, -120960, 2094415, -40042233, ...];
%e A208975 n=8: [1, 8,  4, 48,(-574),8000, -125952, 2208384, -42639617, ...];
%e A208975 n=9: [1, 9,  9, 48, -567,(8082),-129300, 2294784, -44734032, ...];
%e A208975 n=10:[1,10, 15, 50, -555,(8082),-131300, 2357760, -46390320, ...];
%e A208975 n=11:[1,11, 22, 55, -539, 8019,(-132198),2400860, -47665200, ...];
%e A208975 n=12:[1,12, 30, 64, -519, 7908,(-132198),2427120, -48608304, ...];
%e A208975 n=13:[1,13, 39, 78, -494, 7761, -131469,(2439138),-49263084, ...];
%e A208975 n=14:[1,14, 49, 98, -462, 7588, -130151,(2439138),-49667604, ...];
%e A208975 n=15:[1,15, 60,125, -420, 7398, -128360, 2429025,(-49855230), ...];
%e A208975 n=16:[1,16, 72,160, -364, 7200, -126192, 2410432,(-49855230), ...];
%e A208975 where the coefficients in parenthesis demonstrate the property:
%e A208975 [x^n] A(x)^(2*n-1) = [x^n] A(x)^(2*n) for n>=2.
%o A208975 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*A*deriv(x/A^2));polcoeff(A,n)}
%o A208975 for(n=0,25,print1(a(n),", "))
%Y A208975 Cf. A185971, A208961, A000699.
%K A208975 sign
%O A208975 0,3
%A A208975 _Paul D. Hanna_, Mar 03 2012