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EXAMPLE
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G.f.: A(x) = 1 + x - 3*x^2 + 20*x^3 - 189*x^4 + 2232*x^5 - 31130*x^6 +...
Related expansion:
d/dx x/A(x)^2 = 1 - 4*x + 27*x^2 - 248*x^3 + 2830*x^4 - 38232*x^5 +...
Let G(x) be the g.f. of A000699:
G(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 + 38232*x^7 +...
then A(x) = -x/G(-x), or A(x) = 1 + x*A(x) * (x + G(-x))/x^2.
The coefficients in A(x)^n begin:
n=1: [1, 1, -3, 20, -189, 2232, -31130, 497016, -8907885, ...];
n=2: [1, 2, -5, 34, -329, 3966, -56262, 910820, -16509957, ...];
n=3: [1, 3,(-6),43, -429, 5289, -76350, 1253250, -22971165, ...];
n=4: [1, 4,(-6),48, -497, 6276, -92214, 1534560, -28436085, ...];
n=5: [1, 5, -5,(50),-540, 6991, -104555, 1763610, -33031335, ...];
n=6: [1, 6, -3,(50),-564, 7488, -113969, 1948038, -36867735, ...];
n=7: [1, 7, 0, 49,(-574),7812, -120960, 2094415, -40042233, ...];
n=8: [1, 8, 4, 48,(-574),8000, -125952, 2208384, -42639617, ...];
n=9: [1, 9, 9, 48, -567,(8082),-129300, 2294784, -44734032, ...];
n=10:[1,10, 15, 50, -555,(8082),-131300, 2357760, -46390320, ...];
n=11:[1,11, 22, 55, -539, 8019,(-132198),2400860, -47665200, ...];
n=12:[1,12, 30, 64, -519, 7908,(-132198),2427120, -48608304, ...];
n=13:[1,13, 39, 78, -494, 7761, -131469,(2439138),-49263084, ...];
n=14:[1,14, 49, 98, -462, 7588, -130151,(2439138),-49667604, ...];
n=15:[1,15, 60,125, -420, 7398, -128360, 2429025,(-49855230), ...];
n=16:[1,16, 72,160, -364, 7200, -126192, 2410432,(-49855230), ...];
where the coefficients in parenthesis demonstrate the property:
[x^n] A(x)^(2*n-1) = [x^n] A(x)^(2*n) for n>=2.
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