A237646 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).

1, 1, 8, 7, 63, 56, 329, 273, 1736, 1463, 7511, 6048, 32585, 26537, 124440, 97903, 475287, 377384, 1658881, 1281497, 5783960, 4502463, 18825023, 14322560, 61171649, 46849089, 188181672, 141332583, 577889023, 436556440, 1696298665, 1259742225, 4970284200, 3710541975, 14019036535, 10308494560

Offset: 0

Author

Paul D. Hanna, May 03 2014

Keywords

• nonn

Comments

Compare to the g.f. of A195586.

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 7*x^3 + 63*x^4 + 56*x^5 + 329*x^6 + 273*x^7 +...
where
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 +...
Bisections: let A(x) = B(x^2) + x*C(x^2), then:
B(x) = 1 + 8*x + 63*x^2 + 329*x^3 + 1736*x^4 + 7511*x^5 + 32585*x^6 +...
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 +...
Note that C(x)^(1/7) = (1+x+x^2) * C(x^2)^(4/7) is an integer series:
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 +...
Also, C(x) / (1+x+x^2)^3 = A(x)^4:
A(x)^4 = 1 + 4*x + 38*x^2 + 128*x^3 + 817*x^4 + 2536*x^5 + 12890*x^6 +...
Further, C(x)*C(x^2)^3 = A(x)^7:
A(x)^7 = 1 + 7*x + 77*x^2 + 420*x^3 + 2954*x^4 + 13986*x^5 + 78414*x^6 +...
The g.f. may be expressed by the product:
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) *...
		

Crossrefs

Cf. A237647, A237648, A237649, A195586.

Programs

PARI

{A163659(n)=if(n<1, 0, if(n%3, 1, -2)*sigma(2^valuation(n, 2)))}
{a(n)=polcoeff(exp(sum(k=1, n, A163659(k^3)*x^k/k)+x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A237649(n)*x^n/n ), where A237649(n) = A163659(n^3).

G.f. A(x) satisfies:

(1) A(x) = (1+x+x^2) * (1+x^2+x^4)^3 * A(x^2)^4.

(2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(7*4^n).

(3) A(x) / A(-x) = (1+x+x^2) / (1-x+x^2).

Bisections: let A(x) = B(x^2) + x*C(x^2), then

(4) B(x) = (1+x) * C(x).

(5) C(x) = (1+x+x^2)^7 * C(x^2)^4.

(6) A(x) = (1+x+x^2) * C(x^2).

(7) A(x)^7 = C(x) * C(x^2)^3.

(8) A(x)^4 = C(x) / (1+x+x^2)^3.

(9) A(x)^3 = ( C(x)/A(x) - C(x^2)^4/A(x^2)^4 ) / (6*x + 14*x^3 + 6*x^5).