%I #15 Mar 30 2012 18:37:33

%S 1,1,7,199,20026,7296946,10006653574,52756427071846,

%T 1080758244198360481,86574556540356639703921,

%U 27234507698931717202501389871,33749875110161915818408975272861391,165150307912136693948216143106251788630208

%N G.f.: [ Sum_{n>=0} (n+1)*(n+2)/2 * 2^(n*(n-1)) * x^n ]^(1/3).

%C Compare g.f. to: [Sum_{n>=0} (n+1)*(n+2)/2 * x^n ]^(1/3) = 1/(1-x).

%C Conjecture: the characteristic function of a(n) (mod 2) equals (1+x)*(1+x^2)*(1+x^8) * Sum_{n>0} x^(32*A000695(n)), where A000695 is the sums of distinct powers of 4.

%H Paul D. Hanna, <a href="/A202943/b202943.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) == 1 (mod 3).

%e G.f.: A(x) = 1 + x + 7*x^2 + 199*x^3 + 20026*x^4 + 7296946*x^5 +...

%e where

%e A(x)^3 = 1 + 3*x + 6*2^2*x^2 + 10*2^6*x^3 + 15*2^12*x^4 + 21*2^20*x^5 +...

%e more explicitly,

%e A(x)^3 = 1 + 3*x + 24*x^2 + 640*x^3 + 61440*x^4 + 22020096*x^5 +...+ A202944(n)*x^n +...

%e The residues modulo 2 of this sequence begin:

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

%e which forms the characteristic function:

%e (1+x)*(1+x^2)*(1+x^8)*(1 + x^32 + x^128 + x^160 + x^512 + x^544 + x^640 + x^672 +...+ x^(32*A000695(n)) +...).

%o (PARI) {a(n)=polcoeff(sum(m=0,n,(m+1)*(m+2)/2*2^(m*(m-1))*x^m+x*O(x^n))^(1/3),n)}

%Y Cf. A202944, A202942, A202980.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 26 2011