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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). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
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).
EXAMPLE
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) *...
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^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), ", "))
CROSSREFS
Sequence in context: A305255 A358281 A095332 * A120461 A324013 A345013
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 20 2011
EXTENSIONS
Entry and formulas revised by Paul D. Hanna, May 04 2014
STATUS
approved

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Last modified June 19 18:58 EDT 2024. Contains 373507 sequences. (Running on oeis4.)