A317666 G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n) )^n = 1.

1, 2, 7, 48, 590, 10602, 244457, 6767792, 216875258, 7863473864, 317632851912, 14132208327052, 686514289288897, 36154193924315170, 2051928741855927465, 124870207134047889232, 8112089716821244526285, 560396754826502247713090, 41024663835523296400398275, 3172738829903313189522259140, 258493327059457440608140711531

Offset: 0

Author

Paul D. Hanna, Aug 12 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 48*x^3 + 590*x^4 + 10602*x^5 + 244457*x^6 + 6767792*x^7 + 216875258*x^8 + 7863473864*x^9 + 317632851912*x^10 + ...
such that
1 = 1  +  (1/A(x) - (1-x)^2)  +  (1/A(x) - (1-x)^4)^2  +  (1/A(x) - (1-x)^6)^3  +  (1/A(x) - (1-x)^8)^4  +  (1/A(x) - (1-x)^10)^5  +  (1/A(x) - (1-x)^12)^6  +  (1/A(x) - (1-x)^14)^7  +  (1/A(x) - (1-x)^16)^8  + ...
Also,
A(x) = 1  +  (1/A(x) - (1-x)^4)  +  (1/A(x) - (1-x)^6)^2  +  (1/A(x) - (1-x)^8)^3  +  (1/A(x) - (1-x)^10)^4  +  (1/A(x) - (1-x)^12)^5  +  (1/A(x) - (1-x)^14)^6  +  (1/A(x) - (1-x)^16)^7  +  (1/A(x) - (1-x)^18)^8  + ...
RELATED SERIES.
The related series B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+1) )^n begins
B(x) = 1 + x + 3*x^2 + 20*x^3 + 245*x^4 + 4394*x^5 + 101203*x^6 + 2800620*x^7 + 89739208*x^8 + 3253949840*x^9 + 131451064170*x^10 + ...
restated,
B(x) = 1  +  (1/A(x) - (1-x)^3)  +  (1/A(x) - (1-x)^5)^2  +  (1/A(x) - (1-x)^7)^3  +  (1/A(x) - (1-x)^9)^4  +  (1/A(x) - (1-x)^11)^5  +  (1/A(x) - (1-x)^13)^6  +  (1/A(x) - (1-x)^15)^7  +  (1/A(x) - (1-x)^17)^8  + ...
which also equals
B(x) = (1-x)  +  (1/A(x) - (1-x)^4)*(1-x)^2  +  (1/A(x) - (1-x)^6)^2*(1-x)^3  +  (1/A(x) - (1-x)^8)^3*(1-x)^4  +  (1/A(x) - (1-x)^10)^4*(1-x)^5  +  (1/A(x) - (1-x)^12)^5*(1-x)^6  +  (1/A(x) - (1-x)^14)^6*(1-x)^7  +  (1/A(x) - (1-x)^16)^7*(1-x)^8  +  (1/A(x) - (1-x)^18)^8*(1-x)^9  + ...
Compare the above to
1 = (1-x)^2  +  (1/A(x) - (1-x)^4)*(1-x)^4  +  (1/A(x) - (1-x)^6)^2*(1-x)^6  +  (1/A(x) - (1-x)^8)^3*(1-x)^8  +  (1/A(x) - (1-x)^10)^4*(1-x)^10  +  (1/A(x) - (1-x)^12)^5*(1-x)^12  +  (1/A(x) - (1-x)^14)^6*(1-x)^14  +  (1/A(x) - (1-x)^16)^7*(1-x)^16  +  (1/A(x) - (1-x)^18)^8*(1-x)^18  + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A317349, A317667, A317668, A317801.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - (1-x)^(2*m+2) )^m ) )[#A]/2 ); A[n+1]}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n) )^n.

(2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+2) )^n.

(3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+2) )^n * (1-x)^(2*n+2).

(4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+1) )^n ,

then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(2*n+2) )^n * (1-x)^(n+1).

a(n) ~ 2^(n + log(2)/4 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Aug 13 2018