A193021 G.f.: A(x) = 1/(1 - x*(1+x)/(1 - x^2*(1+x)/(1 - x^3*(1+x)/(1 - x^4*(1+x)/(1 - ...))))), a continued fraction.

1, 1, 2, 4, 9, 19, 42, 93, 205, 453, 1003, 2221, 4918, 10892, 24126, 53442, 118384, 262248, 580946, 1286953, 2850965, 6315712, 13991153, 30994539, 68662111, 152107038, 336962513, 746472721, 1653660451, 3663352982, 8115423952, 17978094917

Offset

0,3

Links

Table of n, a(n) for n=0..31.

Formula

G.f.: A(x) = P(x)/Q(x) where

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

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

due to Ramanujan's continued fraction identity.

a(n) ~ c * d^n, where d = 2.2152996327586394990264647692065917932114805328469811... and c = 0.35183326334197478337804661003215013650248042019243949..., d = 1/r, where r = 0.451406204927110926742870001969351624948735236137... is the lowest root of the equation Q(r) = 0. - Vaclav Kotesovec, Aug 25 2017, updated Sep 24 2020

Example

G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 19*x^5 + 42*x^6 +...
which satisfies A(x) = P(x)/Q(x) where
P(x) = 1 - x^2*(1+x)/(1-x) + x^6*(1+x)^2/((1-x)*(1-x^2)) - x^12*(1+x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^20*(1+x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +...
Q(x) = 1 - x*(1+x)/(1-x) + x^4*(1+x)^2/((1-x)*(1-x^2)) - x^9*(1+x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^16*(1+x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +...
Explicitly, the above series begin:
P(x) = 1 - x^2 - 2*x^3 - 2*x^4 - 2*x^5 - x^6 + x^7 + 3*x^8 + 5*x^9 + 7*x^10 + 9*x^11 + 10*x^12 + 9*x^13 + 7*x^14 + 4*x^15 - x^16 - 7*x^17 - 14*x^18 +...
Q(x) = 1 - x - 2*x^2 - 2*x^3 - x^4 + x^5 + 3*x^6 + 5*x^7 + 7*x^8 + 8*x^9 + 7*x^10 + 5*x^11 + 2*x^12 - 3*x^13 - 9*x^14 - 16*x^15 - 24*x^16 - 30*x^17 +...

Mathematica

nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^Range[nmax + 1]*(1+x)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2017 *)

Prog

(PARI) /* As a continued fraction: */
{a(n)=local(A=1+x, CF); CF=1+x; for(k=0, n, CF=1/(1-x^(n-k+1)*(1+x)*CF+x*O(x^n))); A=CF; polcoeff(A, n)}
(PARI) /* By Ramanujan's continued fraction identity: */
{a(n)=local(A=1+x, P, Q); for(i=1, n,
P=sum(m=0, sqrtint(n), x^(m*(m+1))/prod(k=1, m, 1-x^k)*(-1-x+x*O(x^n))^m);
Q=sum(m=0, sqrtint(n), x^(m^2)/prod(k=1, m, 1-x^k)*(-1-x+x*O(x^n))^m); A=P/Q); polcoeff(A, n)}

Crossrefs

Cf. A005169, A192728.

Sequence in context: A127681 A192923 A192673 * A112569 A301706 A289845

Adjacent sequences: A193018 A193019 A193020 * A193022 A193023 A193024

Keyword

nonn

Author

Paul D. Hanna, Jul 14 2011

Status

approved