A338747 G.f. A(x) satisfies: 1 = A(x) - x/(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) ...))))), a continued fraction relation.

1, 1, 0, 1, 1, 6, 17, 79, 330, 1594, 7876, 41433, 226617, 1292848, 7648771, 46853853, 296445440, 1934035905, 12990201995, 89704403890, 636105618633, 4626864097514, 34486708824384, 263162984884732, 2054168834202029, 16388599312054049

Offset

0,6

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 nonnegative 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-2)) * x^n / Product_{k=1..n} (A(x)^k - 1),

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

due to Ramanujan's continued fraction identity.

(2) A(x)^2 = A(x) + x*N(x)/P(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*(n-2)) * x^n / Product_{k=1..n} (A(x)^k - 1).

(3) A(x)^2 = x/(1 - R(x)/Q(x)), where

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

R(x) = Sum_{n>=0} A(x)^(n*(n-4)) * x^n / Product_{k=1..n} (A(x)^k - 1).

(4) A(x) = B(x/A(x)) where B(x) = A(x*B(x)) is the g.f. of A338748.

(5) A(x) = x/Series_Reversion( x*B(x) ) where B(x) is the g.f. of A338748.

Example

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

Crossrefs

Cf. A338748, A338752.

Sequence in context: A047156 A128243 A219294 * A154494 A130278 A024080

Adjacent sequences: A338744 A338745 A338746 * A338748 A338749 A338750

Keyword

nonn

Author

Paul D. Hanna, Nov 06 2020

Status

approved