A187119 - OEIS

%I #5 Mar 30 2012 18:37:26

%S 1,-2,4,-12,36,-140,519,-2632,11776,-82020,426990,-4149112,22719260,

%T -309921456,1487214652,-31477804596,82495148888,-4227929653508,

%U -7155535440434,-767565126075852,-6245080462329816,-194907866415800464

%N  G.f. A(x) satisfies: [x^(m+2)] A(F^m(x)) = 0 for m>0 where F^m(x) denotes the m-th iteration of F(x) = x+x^2 with F^0(x)=x.

%e  G.f.: A(x) = x^2 - 2*x^3 + 4*x^4 - 12*x^5 + 36*x^6 - 140*x^7 +...

%e Let F^n(x) denote the n-th iteration of F(x) = x+x^2 with F^0(x)=x,

%e then the table of coefficients in A(F^n(x)), n>=0, begins:

%e [1, -2, 4, -12, 36, -140, 519, -2632, 11776, ...];

%e [1, 0, -1, -2, -2, -28, -37, -760, -1752, -34632, ...];

%e [1, 2, 0, -6, -18, -64, -284, -1694, -10640, -82576, ...];

%e [1, 4, 7, 0, -46, -232, -1062, -5700, -36354, -268212, ...];

%e [1, 6, 20, 40, 0, -480, -3369, -19988, -126200, -904820, ...];

%e [1, 8, 39, 138, 326, 0, -6309, -56820, -417184, -3091852, ...];

%e [1, 10, 64, 318, 1258, 3492, 0, -100082, -1100188, -9660560, ...];

%e [1, 12, 95, 604, 3242, 14476, 46558, 0, -1859518, -24135624, ...];

%e [1, 14, 132, 1020, 6844, 40348, 202655, 744320, 0, -39597444, ...];

%e [1, 16, 175, 1590, 12750, 92140, 598083, 3354848, 13889080, 0, ...]; ...

%e in which the main diagonal equals all zeros after the initial '1';

%e the lower triangular portion of the above table forms triangle A187115.

%o  (PARI) {ITERATE(F, n, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}

%o {a(n)=local(A=[1]); if(n<2,0,for(i=1, n-1, A=concat(A, 0); A[#A]=-Vec(subst(x^2*Ser(A), x, ITERATE(x+x^2, i, #A)))[#A]); A[n-1])}

%Y  Cf. A187009, A187124, A187115, A135082, A187116, A187117, A187118.

%K sign

%O 2,2

%A _Paul D. Hanna_, Mar 08 2011