OFFSET
0,3
COMMENTS
Note that the continued fraction relation: 1 = F(x) - x*F(x)^k/(F(x) - x*F(x)^k/(F(x) - x*F(x)^k/(F(x) - ...))) holds when F(x) = 1 + x*F(x)^k for a fixed parameter k; this sequence explores the case where the parameter k varies over the positive integers in the continued fraction expression.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) A(x) = P(x)/Q(x), where
P(x) = Sum_{n>=0} A(x)^(n^2) * x^n / Product_{k=1..n} (A(x)^k - 1),
Q(x) = Sum_{n>=0} A(x)^(n*(n-1)) * x^n / Product_{k=1..n} (A(x)^k - 1),
due to Ramanujan's continued fraction identity.
(2) 1/(1-x) = Q(x)/R(x), where
Q(x) = Sum_{n>=0} A(x)^(n*(n-1)) * x^n / Product_{k=1..n} (A(x)^k - 1),
R(x) = Sum_{n>=0} A(x)^(n*(n-2)) * x^n / Product_{k=1..n} (A(x)^k - 1).
(3) A(x)^2 = (x/(1-x)) / (1 - S(x)/R(x)) = (Q(x) - R(x))/(R(x) - S(x)), where
R(x) = Sum_{n>=0} A(x)^(n*(n-2)) * x^n / Product_{k=1..n} (A(x)^k - 1),
S(x) = Sum_{n>=0} A(x)^(n*(n-3)) * x^n / Product_{k=1..n} (A(x)^k - 1).
(4) A(x) = 1/(1 - x*N(x)/P(x)), since Q(x) = P(x) - x*N(x), where
N(x) = Sum_{n>=0} A(x)^(n*(n+1)) * x^n / Product_{k=1..n} (A(x)^k - 1),
P(x) = Sum_{n>=0} A(x)^(n^2) * x^n / Product_{k=1..n} (A(x)^k - 1).
(5) A(x) = B(x*A(x)) where B(x) = A(x/B(x)) is the g.f. of A338748.
(6) A(x) = (1/x)*Series_Reversion( x/B(x) ) where B(x) is the g.f. of A338748.
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 103*x^5 + 513*x^6 + 2762*x^7 + 15796*x^8 + 94869*x^9 + 593701*x^10 + 3850196*x^11 + 25770934*x^12 + ...
where
1 = A(x) - x*A(x)^2/(A(x) - x*A(x)^3/(A(x) - x*A(x)^4/(A(x) - x*A(x)^5/(A(x) - x*A(x)^6/(A(x) - x*A(x)^7/(A(x) - ...)))))).
The g.f. A(x) also satisfies the relation
1/(1-x) = Q(x)/R(x),
where Q(x) and R(x) are defined by
Q(x) = Sum_{n>=0} A(x)^(n*(n-1)) * x^n / Product_{k=1..n} (A(x)^k - 1),
R(x) = Sum_{n>=0} A(x)^(n*(n-2)) * x^n / Product_{k=1..n} (A(x)^k - 1).
Explicitly,
Q(x) = exp(1)*(1 - 5/4*x - 427/288*x^2 - 5995/1152*x^3 - 96742037/4147200*x^4 - 1984683323/16588800*x^5 - 117677276280757/175575859200*x^6 - 2811615436942681/702303436800*x^7 - 127110912548830968887/5056584744960000*x^8 - 3325861987063290233381/20226338979840000*x^9 - 196248007911019260220086647/176211865192366080000*x^10 + ...)
and
R(x) = exp(1)*(1 - 9/4*x - 67/288*x^2 - 1429/384*x^3 - 75160037/4147200*x^4 - 21302869/221184*x^5 - 3866855519605/7023034368*x^6 - 28900078170613/8670412800*x^7 - 106867281402843665687/5056584744960000*x^8 - 939139445622655452611/6742112993280000*x^9 - 1338184786237791005654971/1409694921538928640*x^10 + ...).
RELATED SERIES.
Given B(x) is the g.f. of A338748:
B(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 92*x^6 + 423*x^7 + 2093*x^8 + 10994*x^9 + 60744*x^10 + 350743*x^11 + 2106422*x^12 + ...
then B(x) = A(x/B(x)) and A(x) = B(x*A(x))
where
1 = B(x) - x*B(x)/(B(x) - x*B(x)^2/(B(x) - x*B(x)^3/(B(x) - x*B(x)^4/(B(x) - x*B(x)^5/(B(x) - x*B(x)^6/(B(x) - ...)))))).
PROG
(PARI) {a(n) = my(A=[1], CF=1); for(i=1, n, A=concat(A, 0); for(i=1, #A, CF = Ser(A) - (Ser(A)^(#A-i+2)*x)/CF ); A[#A] = -polcoeff(CF, #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 06 2020
STATUS
approved