A338748 G.f. A(x) satisfies: 1 = A(x) - x*A(x)/(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) - ...))))), a continued fraction relation.

1, 1, 1, 2, 6, 22, 92, 423, 2093, 10994, 60744, 350743, 2106422, 13110304, 84330164, 559367278, 3819233961, 26802388190, 193080823079, 1426252354150, 10792528835886, 83587157097544, 662060553448763, 5358900630188358, 44296806348364981

Offset

0,4

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*(n-1)) * x^n / Product_{k=1..n} (A(x)^k - 1),

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

due to Ramanujan's continued fraction identity.

(2) A(x) = 1 + x*N(x)/P(x), where

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

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

(3) A(x) = x/(1 - R(x)/Q(x)), where

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

R(x) = Sum_{n>=0} A(x)^(n*(n-3)) * x^n / Product_{k=1..n} (A(x)^k - 1).

(4) A(x) = 1 + x/(1 - x*M(x)/N(x)), where

M(x) = Sum_{n>=0} A(x)^(n*(n+1)) * x^n / Product_{k=1..n} (A(x)^k - 1),

N(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 A338747.

(6) A(x) = (1/x)*Series_Reversion( x/B(x) ) where B(x) is the g.f. of A338747.

Example

G.f.: A(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 + ...
where
1 = A(x) - x*A(x)/(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) - ...)))))).
RELATED SERIES.
Given B(x) is the g.f. of A338747:
B(x) = 1 + x + x^3 + x^4 + 6*x^5 + 17*x^6 + 79*x^7 + 330*x^8 + 1594*x^9 + 7876*x^10 + 41433*x^11 + 226617*x^12 + ...
then B(x) = A(x/B(x)) and A(x) = B(x*A(x))
where
1 = B(x) - x/(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) - ...)))))).

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+1)*x)/CF ); A[#A] = -polcoeff(CF, #A-1) ); H=Ser(A); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A338747, A338752.

Sequence in context: A001181 A130579 A279570 * A107945 A279571 A014330

Adjacent sequences: A338745 A338746 A338747 * A338749 A338750 A338751

Keyword

nonn

Author

Paul D. Hanna, Nov 06 2020

Status

approved