A168365 - OEIS

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

%S 1,1,2,12,100,1070,14116,222614,4092964,86058372,2038072080,

%T 53704876742,1558947707912,49437387429404,1700708685654304,

%U 63087246457132054,2510333378920658240,106666494341060615560,4820550659164606795712

%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_0: [(1), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,...]

%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, (86058372), ...]; ...

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

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

%Y Cf. A168362, A168366.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Nov 24 2009