A192405 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n+1) * A(x)^n/(1 - x*A(x)^(2*n)).

1, 0, 1, 2, 4, 11, 33, 99, 310, 1016, 3413, 11682, 40751, 144476, 519013, 1886311, 6928012, 25684055, 96020957, 361742039, 1372442092, 5241062187, 20136335035, 77806111700, 302259125863, 1180207733657, 4630733662020, 18254415188073, 72283753111667

Offset

0,4

Comments

Related q-series identity:

Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)); here q=A(x), y=x, z=x.

Links

Table of n, a(n) for n=0..28.

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n+1)*A(x)^(2*n-1)/(1 - x*A(x)^(2*n-1)).

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n+1)*A(x)^(n*(n+1)/2) * Sum_{k=0..n-1} A(x)^(-k*(k+1)/2).

Equals the antidiagonal sums of square array A192404.

Example

G.f.: A(x) = 1 + x^2 + 2*x^3 + 4*x^4 + 11*x^5 + 33*x^6 + 99*x^7 +...
which satisfies the following relations:
A(x) = 1 + x^2*A(x)/(1-x*A(x)^2) + x^3*A(x)^2/(1-x*A(x)^4) + x^4*A(x)^3/(1-x*A(x)^6) +...
A(x) = 1 + x^2*A(x)/(1-x*A(x)) + x^3*A(x)^3/(1-x*A(x)^3) + x^4*A(x)^5/(1-x*A(x)^5) +...
A(x) = 1 + x^2*A(x) + x^3*A(x)^3*(1 + 1/A(x)) + x^4*A(x)^6*(1 + 1/A(x) + 1/A(x)^3) + x^5*A(x)^10*(1 + 1/A(x) + 1/A(x)^3 + 1/A(x)^6) +...

Prog

(PARI) {a(n)=local(A=1+x^2); for(i=1, n, A=1+x*sum(m=1, n, x^m*A^m/(1-x*A^(2*m)+x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x^2); for(i=1, n, A=1+x*sum(m=1, n, x^m*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^(m+1)*A^(m*(m+1)/2)*sum(k=0, m-1, (A+x*O(x^n))^(-k*(k+1)/2) ) ) ); polcoeff(A, n)}

Crossrefs

Cf. A192404, A192406, A192399, A192401.

Sequence in context: A123439 A026164 A025191 * A035354 A220902 A249945

Adjacent sequences: A192402 A192403 A192404 * A192406 A192407 A192408

Keyword

nonn

Author

Paul D. Hanna, Jun 30 2011

Status

approved