OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1200
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = x * Product_{n>=0} (1 + A(x)^(2^n))^(2^n).
(2) A(x) = Series_Reversion( x/Product_{n>=0} (1 + x^(2^n))^(2^n) ).
(3) A( x^2*sqrt((1+x)/B(x)) ) = x^2 where A(x/B(x)) = x and B(x) = (1+x)*B(x^2)^2 is the g.f. of A073707.
(4) A(x)^2 = A( sqrt(x*A(x)^3 + x*A(x)^4) ).
a(n) ~ c * d^n / n^(3/2), where d = 5.164144165326872409478949107112602669075360713205313608870833506... and c = 0.097143352580345193472769014811303851564910257997894161469730732... - Vaclav Kotesovec, May 19 2026
EXAMPLE
G.f.: A(x) = x + x^2 + 3*x^3 + 9*x^4 + 34*x^5 + 131*x^6 + 536*x^7 + 2249*x^8 + 9701*x^9 + 42622*x^10 + 190308*x^11 + 860603*x^12 + ...
where A(x) = x * (1 + A(x)) * (1 + A(x)^2)^2 * (1 + A(x)^4)^4 * (1 + A(x)^8)^8 * ... * (1 + A(x)^(2^n))^(2^n) * ...
RELATED SERIES.
Let B(x) be the g.f. of A073707, which begins
B(x) = 1 + x + 2*x^2 + 2*x^3 + 5*x^4 + 5*x^5 + 8*x^6 + 8*x^7 + 18*x^8 + 18*x^9 + 28*x^10 + 28*x^11 + 50*x^12 + ... + A073707(n)*x^n + ...
and B(x)^2 = 1 + 2*x + 5*x^2 + 8*x^3 + 18*x^4 + 28*x^5 + 50*x^6 + 72*x^7 + 129*x^8 + ... + A073708(n)*x^n + ...
then A(x/B(x)) = x and B(x) = (1+x)*B(x^2)^2.
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.192566184665525627373764429720275579970799922502815... where t = 1/(3*Product_{n>=0} (1 + 1/3^(2^n))^(2^n) ).
A(t) = 1/4 at t = 0.174399995931454056980340659148621585409988754104063... where t = 1/(4*Product_{n>=0} (1 + 1/4^(2^n))^(2^n) ).
A(t) = 1/5 at t = 0.153107305379131511461490934290043705174922409899435... where t = 1/(5*Product_{n>=0} (1 + 1/5^(2^n))^(2^n) ).
A(1/6) = 0.229703382795427621236927298529391027047646343128809...
A(1/7) = 0.180609457989966641735006253305482970425376856270621...
A(1/8) = 0.150772380853740786368498741228317383164086142585670...
A(1/9) = 0.129979937696592592485000152585700658884791899556917...
A(1/10) = 0.11446455123152402040885008686574591650787223523666...
PROG
(PARI) {a(n) = my(A = serreverse(x/prod(m=0, #binary(n), (1 + x^(2^m) +x*O(x^n))^(2^m)))); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 17 2025
STATUS
approved