%I #5 Mar 15 2022 08:20:45

%S 1,1,-3,7,-25,99,-424,1906,-8874,42412,-206878,1025718,-5154020,

%T 26188190,-134327493,694610923,-3617170909,18952557141,-99844769274,

%U 528543587562,-2810053368606,14998219677082,-80333361944524,431662921085376,-2326299337924086

%N G.f. A(x) satisfies: A(x)^3 = (1-x) * (A(x) + x)^2.

%F G.f. A(x) satisfies:

%F (1) A(x) = sqrt( A(x)^3/(1-x) ) - x.

%F (2) A(x)^3 = (1-x) * (A(x) + x)^2.

%F (3) A( x*(1+x)^2/(1 + x*(1+x)^2) ) = (1+x)^2/(1 + x*(1+x)^2).

%F (4) A(x) = x / Series_Reversion( x*(1+x)^2/(1 + x*(1+x)^2) ).

%F (5) Sum_{k=0..n} [x^k] A(x)^n = (-1)^(n-1) * 2, for n >= 1.

%e G.f.: A(x) = 1 + x - 3*x^2 + 7*x^3 - 25*x^4 + 99*x^5 - 424*x^6 + 1906*x^7 - 8874*x^8 + 42412*x^9 - 206878*x^10 + ...

%e where A(x)^3 equals (1-x)*(A(x) + x)^2, as can be seen from the following power series expansions:

%e A(x)^3 = 1 + 3*x - 6*x^2 + 4*x^3 - 15*x^4 + 69*x^5 - 309*x^6 + 1425*x^7 - 6741*x^8 + 32575*x^9 + ...

%e (A(x) + x)^2 = 1 + 4*x - 2*x^2 + 2*x^3 - 13*x^4 + 56*x^5 - 253*x^6 + 1172*x^7 - 5569*x^8 + 27006*x^9 + ...

%e Related table.

%e Another defining property of the g.f. A(x) is illustrated here.

%e The table of coefficients of x^k in A(x)^n begins:

%e n=1: [1, 1, -3, 7, -25, 99, -424, 1906, -8874, ...];

%e n=2: [1, 2, -5, 8, -27, 106, -451, 2020, -9381, ...];

%e n=3: [1, 3, -6, 4, -15, 69, -309, 1425, -6741, ...];

%e n=4: [1, 4, -6, -4, 3, 24, -144, 744, -3747, ...];

%e n=5: [1, 5, -5, -15, 20, -4, -35, 265, -1580, ...];

%e n=6: [1, 6, -3, -28, 30, 0, -8, 48, -447, ...];

%e n=7: [1, 7, 0, -42, 28, 42, -49, 15, -63, ...];

%e n=8: [1, 8, 4, -56, 10, 120, -116, 24, 3, ...]; ...

%e in which the partial sum of row n up to column n equals (-1)^(n-1)*2, as illustrated by:

%e n=1: 2 = 1 + 1;

%e n=2: -2 = 1 + 2 + -5;

%e n=3: 2 = 1 + 3 + -6 + 4;

%e n=4: -2 = 1 + 4 + -6 + -4 + 3;

%e n=5: 2 = 1 + 5 + -5 + -15 + 20 + -4;

%e n=6: -2 = 1 + 6 + -3 + -28 + 30 + 0 + -8;

%e n=7: 2 = 1 + 7 + 0 + -42 + 28 + 42 + -49 + 15;

%e n=8: -2 = 1 + 8 + 4 + -56 + 10 + 120 + -116 + 24 + 3;

%e ...

%o (PARI) {a(n) = polcoeff( x/serreverse( x*(1+x)^2/(1 + x*(1+x)^2 +x^2*O(x^n)) ),n)}

%o for(n=0,30,print1(a(n),", "))

%K sign

%O 0,3

%A _Paul D. Hanna_, Mar 14 2022