OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..200
FORMULA
Let A = g.f. A(x), then A satisfies:
(1) A = Sum_{n>=0} x^n*(1+xA)^n*Product_{k=1..n} (1-(1+xA)*x^(2k-1))/(1-(1+xA)*x^(2k)) due to a q-series identity.
(2) A = 1/(1- x*(1+xA)/(1- (x^2-x)*(1+xA)/(1- x^3*(1+xA)/(1- (x^4-x^2)*(1+xA)/(1- x^5*(1+xA)/(1- (x^6-x^3)*(1+xA)/(1- ...))))))) due to a continued fraction of a partial theta function.
a(n) ~ c * d^n / n^(3/2), where d = 2.01316244..., c = 3.801615... . - Vaclav Kotesovec, Aug 25 2014
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 15*x^7 +...
Let A = A(x) then A satisfies:
A = 1 + (x + x^2*A) + (x^3 + 2*x^4*A + x^5*A^2) + (x^6 + 3*x^7*A + 3*x^8*A^2 + x^9*A^3) + (x^10 + 4*x^11*A + 6*x^12*A^2 + 4*x^13*A^3 + x^14*A^4) +...
in which all the binomial coefficients appear in succession.
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, (1+x*A+x*O(x^n))^m*x^(m*(m+1)/2))); polcoeff(A, n)}
(PARI) /* q-series expansion: */ {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*(1+x*A)^m*prod(k=1, m, (1-(1+x*A)*x^(2*k-1))/(1-(1+x*A)*x^(2*k)+x*O(x^n))))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 09 2010
STATUS
approved