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A195586 G.f.: exp( Sum_{n>=1} A163659(n^2)*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).

%I A195586 #29 May 04 2014 13:01:21
%S A195586 1,1,4,3,15,12,37,25,100,75,219,144,501,357,972,615,1995,1380,3665,
%T A195586 2285,7052,4767,12255,7488,22305,14817,37524,22707,65775,43068,106837,
%U A195586 63769,180436,116667,286251,169584,471173,301589,729404,427815,1169211,741396,1778545,1037149
%N A195586 G.f.: exp( Sum_{n>=1} A163659(n^2)*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).
%H A195586 Paul D. Hanna, <a href="/A195586/b195586.txt">Table of n, a(n) for n = 0..1000</a>
%F A195586 G.f.: exp( Sum_{n>=1} A195587(n)*x^n/n ), where A195587(n) = A163659(n^2).
%F A195586 G.f. A(x) satisfies:
%F A195586 (1) A(x) = (1+x+x^2) * (1+x^2+x^4) * A(x^2)^2.
%F A195586 (2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(3*2^n).
%F A195586 (3) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).
%F A195586 Bisections: let A(x) = B(x^2) + x*C(x^2), then
%F A195586 (4) B(x) = (1+x) * C(x).
%F A195586 (5) C(x) = (1+x+x^2)^3 * C(x^2)^2.
%F A195586 (6) A(x) = (1+x+x^2) * C(x^2).
%F A195586 (7) A(x)^3 = C(x) * C(x^2).
%F A195586 (8) A(x)^2 = C(x) / (1+x+x^2).
%F A195586 (9) A(x) = ( C(x)/A(x) - C(x^2)^2/A(x^2)^2 ) / (2*x).
%e A195586 G.f.: A(x) = 1 + x + 4*x^2 + 3*x^3 + 15*x^4 + 12*x^5 + 37*x^6 + 25*x^7 +...
%e A195586 where
%e A195586 log(A(x)) = x + 7*x^2/2 - 2*x^3/3 + 31*x^4/4 + x^5/5 - 14*x^6/6 + x^7/7 + 127*x^8/8 +...+ A195587(n)*x^n/n +...
%e A195586 Let C(x) be the odd bisection of g.f. A(x):
%e A195586 C(x) = 1 + 3*x + 12*x^2 + 25*x^3 + 75*x^4 + 144*x^5 + 357*x^6 + 615*x^7 + 1380*x^8 + 2285*x^9 + 4767*x^10 + 7488*x^11 + 14817*x^12 +...+ A237650(n)*x^n +...
%e A195586 then C(x) equals the cube of an integer series:
%e A195586 C(x)^(1/3) = 1 + x + 3*x^2 + 2*x^3 + 9*x^4 + 7*x^5 + 17*x^6 + 10*x^7 + 41*x^8 + 31*x^9 + 75*x^10 + 44*x^11 + 150*x^12 +...+ A237651(n)*x^n +...
%e A195586 which equals A(x)/C(x^2)^(1/3).
%e A195586 The g.f. may be expressed by the product:
%e A195586 A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * (1+x^4+x^8)^6 * (1+x^8+x^16)^12 * (1+x^16+x^32)^24 *...* (1 + x^(2*2^n) + x^(4*2^n))^(3*2^n) *...
%o A195586 (PARI) {A163659(n)=if(n<1,0,if(n%3,1,-2)*sigma(2^valuation(n,2)))}
%o A195586 {a(n)=polcoeff(exp(sum(k=1, n, A163659(k^2)*x^k/k)+x*O(x^n)), n)}
%o A195586 for(n=0,50,print1(a(n),", "))
%o A195586 (PARI) /* G.f.: A(x) = (1+x+x^2) * (1+x^2+x^4) * A(x^2)^2: */
%o A195586 {a(n)=local(A=1+x); for(i=1, #binary(n), A=(1+x+x^2)*(1+x^2+x^4)*subst(A^2, x, x^2) +x*O(x^n)); polcoeff(A, n)}
%o A195586 for(n=0, 50, print1(a(n), ", "))
%o A195586 (PARI) /* G.f.: (1+x+x^2) * Product_{n>=0} (1 + x^(2*2^n) + x^(4*2^n))^(3*2^n): */
%o A195586 {a(n)=local(A=1+x); A=(1+x+x^2)*prod(k=0, #binary(n), (1+x^(2*2^k)+x^(4*2^k)+x*O(x^n))^(3*2^k)); polcoeff(A, n)}
%o A195586 for(n=0, 50, print1(a(n), ", "))
%Y A195586 Cf. A195587, A163658, A163659, A237650, A002487; variant: A237646.
%K A195586 nonn
%O A195586 0,3
%A A195586 _Paul D. Hanna_, Sep 20 2011
%E A195586 Entry and formulas revised by _Paul D. Hanna_, May 04 2014