A168366 - OEIS

%I #2 Mar 30 2012 18:37:20

%S 1,3,12,75,650,7238,98728,1597689,29965770,639867250,15337926824,

%T 408041226684,11934806334412,380779156844728,13163263492873152,

%U 490208180677163141,19568027053121979366,833594881998712829460

%N a(n) = coefficient of x^n in the (n+1)-th iteration of Sum_{k>=0} x^(2^k), n>=1.

%e Let F(x) = x + x^2 + x^4 + x^8 + x^16 +...+ x^(2^n) +...

%e and define F_{n}(x) = F_{n-1}(x*F(x)) as the n-th iteration of F(x)

%e then the coefficients in the successive iterations of F(x) begin:

%e F_1: [1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, ...];

%e F_2: [(1), 2, 2, 3, 6, 8, 8, 16, 22, 40, 80, 146, 240, 356,...];

%e F_3: [1, (3), 6, 12, 28, 68, 152, 330, 722, 1614, 3680, 8810, ...];

%e F_4: [1, 4, (12), 34, 100, 310, 972, 3012, 9190, 27826, 84192,...];

%e F_5: [1, 5, 20, (75), 280, 1070, 4176, 16449, 64710, 253166, ...];

%e F_6: [1, 6, 30, 141, (650), 3010, 14116, 66972, 319788, 1529230,...];

%e F_7: [1, 7, 42, 238, 1316, (7238), 39984, 222614, 1248304,...];

%e F_8: [1, 8, 56, 372, 2408, 15428, (98728), 634040, 4092964,...];

%e F_9: [1, 9, 72, 549, 4080, 29940, 218688, (1597689), 11704548,...];

%e F_10:[1, 10, 90, 775, 6510, 53940, 443952, 3646168, (29965770),...]; ...

%e where the coefficients in parenthesis form the initial terms of this sequence.

%o (PARI) {a(n)=local(G=x,S2=sum(m=0,#binary(n)+1,x^(2^m))+x*O(x^n)); for(i=1,n+1,G=subst(S2,x,G));polcoeff(G,n)}

%Y Cf. A168362, A168365.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 24 2009