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%I #5 Mar 16 2022 16:32:48
%S 1,2,-6,22,-115,675,-4268,28328,-194664,1372984,-9882190,72293094,
%T -535946436,4017628380,-30402186474,231924537486,-1781685227700,
%U 13771456884192,-107024198084814,835752089556270,-6554617760700801,51606615762239313
%N G.f. A(x) satisfies: A(x)^4 = (1-x) * (A(x) + x)^3.
%F G.f. A(x) satisfies:
%F (1) A(x) = ( A(x)^4/(1-x) )^(1/3) - x.
%F (2) A(x)^4 = (1-x) * (A(x) + x)^3.
%F (3) A( x*(1+x)^3/(1 + x*(1+x)^3) ) = (1+x)^3/(1 + x*(1+x)^3).
%F (4) A(x) = x / Series_Reversion( x*(1+x)^3/(1 + x*(1+x)^3) ).
%F (5) Sum_{k=0..n} [x^k] A(x)^n = (-1)^(n-1) * 3, for n >= 1.
%e G.f.: A(x) = 1 + 2*x - 6*x^2 + 22*x^3 - 115*x^4 + 675*x^5 - 4268*x^6 + 28328*x^7 - 194664*x^8 + ...
%e where A(x)^4 equals (1-x)*(A(x) + x)^3, as can be seen from the following power series expansions:
%e A(x)^4 = 1 + 8*x + 0*x^2 - 24*x^3 + 12*x^4 + 84*x^5 - 840*x^6 + 6816*x^7 - 52530*x^8 + ...
%e (A(x) + x)^3 = 1 + 9*x + 9*x^2 - 15*x^3 - 3*x^4 + 81*x^5 - 759*x^6 + 6057*x^7 - 46473*x^8 + ...
%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, 2, -6, 22, -115, 675, -4268, 28328, ...];
%e n=2: [1, 4, -8, 20, -106, 626, -3972, 26424, ...];
%e n=3: [1, 6, -6, 2, -45, 333, -2292, 15948, ...];
%e n=4: [1, 8, 0, -24, 12, 84, -840, 6816, ...];
%e n=5: [1, 10, 10, -50, 25, 7, -140, 1840, ...];
%e n=6: [1, 12, 24, -68, -30, 102, -44, 216, ...];
%e n=7: [1, 14, 42, -70, -161, 273, -84, -12, ...]; ...
%e in which the partial sum of row n up to column n equals (-1)^(n-1)*3, as illustrated by:
%e n=1: 3 = 1 + 2;
%e n=2: -3 = 1 + 4 + -8;
%e n=3: 3 = 1 + 6 + -6 + 2;
%e n=4: -3 = 1 + 8 + 0 + -24 + 12;
%e n=5: 3 = 1 + 10 + 10 + -50 + 25 + 7;
%e n=6: -3 = 1 + 12 + 24 + -68 + -30 + 102 + -44;
%e n=7: 3 = 1 + 14 + 42 + -70 + -161 + 273 + -84 + -12;
%e ...
%o (PARI) {a(n) = polcoeff( x/serreverse( x*(1+x)^3/(1 + x*(1+x)^3 +x^2*O(x^n)) ),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A352385, A352414, A352415.
%K sign
%O 0,2
%A _Paul D. Hanna_, Mar 15 2022
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