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A190790 G.f. satisfies: A(x) = 1 + Sum_{n>=1} q^(2n-1)/(1 - q^(2n-1)) where q = x*A(x). 2
1, 1, 2, 6, 18, 58, 198, 696, 2506, 9205, 34344, 129792, 495834, 1911640, 7428444, 29064650, 114404410, 452719183, 1799994588, 7187148262, 28807364008, 115865980972, 467497031164, 1891710323324, 7675031497682, 31215088847239 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..25.

FORMULA

G.f. A(x) satisfies:

* A(x) = 1 + Sum_{n>=1} q^(n*(n+1)/2)/(1 - q^n), where q = x*A(x);

* A(x) = 1 + Sum_{n>=1} q^n/(1 - q^(2n)), where q = x*A(x);

* A(x) = 1 + Sum_{n>=1} A001227(n)*x^n*A(x)^n, where A001227(n) = number of odd divisors of n.

Let D(x) = 1 + Sum_{n>=1} A001227(n)*x^n, then

* A(x) = D(x*A(x)) and D(x) = A(x/D(x));

* A(x) = (1/x)*Series_Reversion(x/D(x));

* a(n) = [x^n] D(x)^(n+1)/(n+1), the coefficient of x^n in D(x)^(n+1)/(n+1) for n>=0.

EXAMPLE

G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 18*x^4 + 58*x^5 + 198*x^6 +...

Let q = x*A(x), then the g.f. A(x) satisfies the following series:

* A(x) = 1 + q/(1 - q) + q^3/(1 - q^3) + q^5/(1 - q^5) + q^7/(1 - q^7) +...

* A(x) = 1 + q/(1 - q) + q^3/(1 - q^2) + q^6/(1 - q^3) + q^10/(1 - q^4) +...

* A(x) = 1 + q/(1 - q^2) + q^2/(1 - q^4) + q^3/(1 - q^6) + q^4/(1 - q^8) +...

* A(x) = 1 + q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 +...+ A001227(n)*q^n +...

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, (x*A)^(2*m-1)/(1-(x*A)^(2*m-1)+x*O(x^n)))); polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, (x*A)^(m*(m+1)/2)/(1-(x*A)^m+x*O(x^n)))); polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, (x*A)^m/(1-(x*A)^(2*m)+x*O(x^n)))); polcoeff(A, n)}

(PARI) {a(n)=local(D=1+sum(m=1, n, sumdiv(m, d, d%2)*x^m)+x*O(x^n)); polcoeff(1/x*serreverse(x/D), n)}

(PARI) {a(n)=local(D=1+sum(m=1, n, sumdiv(m, d, d%2)*x^m)+x*O(x^n)); polcoeff(D^(n+1)/(n+1), n)}

CROSSREFS

Sequence in context: A293067 A085139 A150041 * A150042 A036675 A227373

Adjacent sequences:  A190787 A190788 A190789 * A190791 A190792 A190793

KEYWORD

nonn

AUTHOR

Paul D. Hanna, May 20 2011

STATUS

approved

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Last modified June 6 18:59 EDT 2020. Contains 334832 sequences. (Running on oeis4.)