_Paul D. Hanna (pauldhanna(AT)juno.com), _, Mar 08 2011

reviewed

approved

proposed

reviewed

allocated G.f. A(x) satisfies: [x^(n+3)] A(F^n(x)) = 0 for Paul Dn>0 where F^n(x) denotes the n-th iteration of F(x) = x+x^2 with F^0(x)=x. Hanna

1, -3, 6, -18, 48, -195, 549, -3465, 7452, -112707, -5994, -6866904, -25659292, -700243362, -5594278734, -106900155574, -1284177510456, -22692117042216, -348993455353854, -6343625959542180, -114598750263323292

3,2

G.f.: A(x) = x^3 - 3*x^4 + 6*x^5 - 18*x^6 + 48*x^7 - 195*x^8 +...

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

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

[1, -3, 6, -18, 48, -195, 549, -3465, 7452, -112707, ...];

[1, 0, -3, -5, -12, -72, -333, -2568, -16782, -153204, ...];

[1, 3, 0, -19, -72, -261, -1276, -8079, -58932, -486635, ...];

[1, 6, 15, 0, -174, -1047, -5256, -29676, -202908, -1625427, ...];

[1, 9, 42, 112, 0, -2109, -17211, -112371, -753606, -5711283, ...];

[1, 12, 81, 377, 1128, 0, -31633, -324600, -2614344, -20650886, ...];

[1, 15, 132, 855, 4248, 14373, 0, -564081, -6957390, -66648777, ...];

[1, 18, 195, 1606, 10758, 58269, 221952, 0, -11639502, -167467539,..];

[1, 21, 270, 2690, 22416, 159633, 947117, 4029915, 0, -272551739,...];

[1, 24, 357, 4167, 41340, 359616, 2750067, 17848872, 84135510, 0,...]; ...

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

the lower triangular portion of the above table forms triangle A187124.

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

{a(n)=local(A=[1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(subst(x^3*Ser(A), x, ITERATE(x+x^2, i, #A)))[#A]); A[n]}

Cf. A187009, A187119, A187120, A135083, A187121, A187122, A187123.

allocated

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Mar 08 2011

approved

proposed

allocated for Paul D. Hanna

allocated

approved