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).

1, 1, 4, 3, 15, 12, 37, 25, 100, 75, 219, 144, 501, 357, 972, 615, 1995, 1380, 3665, 2285, 7052, 4767, 12255, 7488, 22305, 14817, 37524, 22707, 65775, 43068, 106837, 63769, 180436, 116667, 286251, 169584, 471173, 301589, 729404, 427815, 1169211, 741396, 1778545, 1037149

Offset: 0

Author

Paul D. Hanna, Sep 20 2011

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 3*x^3 + 15*x^4 + 12*x^5 + 37*x^6 + 25*x^7 +...
where
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 +...
Let C(x) be the odd bisection of g.f. A(x):
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 +...
then C(x) equals the cube of an integer series:
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 +...
which equals A(x)/C(x^2)^(1/3).
The g.f. may be expressed by the product:
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) *...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A195587, A163658, A163659, A237650, A002487; variant: A237646.

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^2)*x^k/k)+x*O(x^n)), n)}
for(n=0,50,print1(a(n),", "))

PARI

/* G.f.: A(x) = (1+x+x^2) * (1+x^2+x^4) * A(x^2)^2: */
{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)}
for(n=0, 50, print1(a(n), ", "))

PARI

/* G.f.: (1+x+x^2) * Product_{n>=0} (1 + x^(2*2^n) + x^(4*2^n))^(3*2^n): */
{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)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f.: exp( Sum_{n>=1} A195587(n)*x^n/n ), where A195587(n) = A163659(n^2).

G.f. A(x) satisfies:

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

(2) A(x) = (1+x+x^2) * Product_{n>=0} ( 1 + x^(2*2^n) + x^(4*2^n) )^(3*2^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)^3 * C(x^2)^2.

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

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

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

(9) A(x) = ( C(x)/A(x) - C(x^2)^2/A(x^2)^2 ) / (2*x).

Extensions

Entry and formulas revised by Paul D. Hanna, May 04 2014