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%I #9 May 03 2014 20:35:56
%S 1,1,8,7,63,56,329,273,1736,1463,7511,6048,32585,26537,124440,97903,
%T 475287,377384,1658881,1281497,5783960,4502463,18825023,14322560,
%U 61171649,46849089,188181672,141332583,577889023,436556440,1696298665,1259742225,4970284200,3710541975,14019036535,10308494560
%N G.f.: exp( Sum_{n>=1} A163659(n^3)*x^n/n ), where x*exp(Sum_{n>=1} A163659(n)*x^n/n) = S(x) is the g.f. of Stern's diatomic series (A002487).
%C Compare to the g.f. of A195586.
%F G.f.: exp( Sum_{n>=1} A237649(n)*x^n/n ), where A237649(n) = A163659(n^3).
%F G.f. A(x) satisfies:
%F (1) A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * A(x^2)^4.
%F (2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(7*4^n).
%F (3) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).
%F Bisections: let A(x) = B(x^2) + x*C(x^2), then
%F (4) B(x) = (1+x) * C(x).
%F (5) C(x) = (1+x+x^2)^7 * C(x^2)^4.
%F (6) A(x) = (1+x+x^2) * C(x^2).
%F (7) A(x)^7 = C(x) * C(x^2)^3.
%F (8) A(x)^4 = C(x) / (1+x+x^2)^3.
%F (9) A(x)^3 = ( C(x)/A(x) - C(x^2)^4/A(x^2)^4 ) / (6*x + 14*x^3 + 6*x^5).
%e G.f.: A(x) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 +...
%e where
%e log(A(x)) = x + 15*x^2/2 - 2*x^3/3 + 127*x^4/4 + x^5/5 - 30*x^6/6 + x^7/7 + 1023*x^8/8 +...+ A237649(n)*x^n/n +...
%e Bisections: let A(x) = B(x^2) + x*C(x^2), then:
%e B(x) = 1 + 8*x + 63*x^2 + 329*x^3 + 1736*x^4 + 7511*x^5 + 32585*x^6 +...
%e C(x) = 1 + 7*x + 56*x^2 + 273*x^3 + 1463*x^4 + 6048*x^5 + 26537*x^6 + 97903*x^7 + 377384*x^8 + 1281497*x^9 + 4502463*x^10 +...+ A237647(n)*x^n +...
%e Note that C(x)^(1/7) = (1+x+x^2) * C(x^2)^(4/7) is an integer series:
%e C(x)^(1/7) = 1 + x + 5*x^2 + 4*x^3 + 30*x^4 + 26*x^5 + 106*x^6 + 80*x^7 + 459*x^8 + 379*x^9 + 1451*x^10 + 1072*x^11 + 5210*x^12 +...+ A237648(n)*x^n +...
%e Also, C(x) / (1+x+x^2)^3 = A(x)^4:
%e A(x)^4 = 1 + 4*x + 38*x^2 + 128*x^3 + 817*x^4 + 2536*x^5 + 12890*x^6 +...
%e Further, C(x)*C(x^2)^3 = A(x)^7:
%e A(x)^7 = 1 + 7*x + 77*x^2 + 420*x^3 + 2954*x^4 + 13986*x^5 + 78414*x^6 +...
%e The g.f. may be expressed by the product:
%e A(x) = (1+x+x^2) * (1+x^2+x^4)^7 * (1+x^4+x^8)^28 * (1+x^8+x^16)^112 * (1+x^16+x^32)^448 *...* (1 + x^(2*2^n) + x^(4*2^n))^(7*4^n) *...
%o (PARI) {A163659(n)=if(n<1, 0, if(n%3, 1, -2)*sigma(2^valuation(n, 2)))}
%o {a(n)=polcoeff(exp(sum(k=1, n, A163659(k^3)*x^k/k)+x*O(x^n)), n)}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A237647, A237648, A237649, A195586.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 03 2014
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