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

0,3

Comments

Compare to the g.f. of A195586.

Links

Table of n, a(n) for n=0..35.

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).

Example

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) *...

Prog

(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), ", "))

Crossrefs

Cf. A237647, A237648, A237649, A195586.

Sequence in context: A138809 A286460 A317231 * A288188 A038285 A354546

Adjacent sequences: A237643 A237644 A237645 * A237647 A237648 A237649

Keyword

nonn

Author

Paul D. Hanna, May 03 2014

Status

approved