A192737 G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x*A(x)^2/(1 - x*A(x)^3/(1 - x*A(x)^4/(1 - ...))))), a recursive continued fraction.

1, 1, 3, 13, 68, 399, 2531, 16994, 119199, 865849, 6474177, 49616016, 388484212, 3100311228, 25172981053, 207665895001, 1738775327319, 14764815028481, 127076945426555, 1108103873824072, 9787004793441886, 87539719110388691

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction.

Formula

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

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

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

due to Ramanujan's continued fraction identity.

Example

G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2531*x^6 +...
The g.f. A = A(x) satisfies A = P(x)/Q(x) where
P(x) = 1 - x*A^2/(1-A) + x^2*A^6/((1-A)*(1-A^2)) - x^3*A^12/((1-A)*(1-A^2)*(1-A^3)) + x^4*A^20/((1-A)*(1-A^2)*(1-A^3)*(1-A^4)) -+...
Q(x) = 1 - x*A/(1-A) + x^2*A^4/((1-A)*(1-A^2)) - x^3*A^9/((1-A)*(1-A^2)*(1-A^3)) + x^4*A^16/((1-A)*(1-A^2)*(1-A^3)*(1-A^4)) -+...
Explicitly, the above series begin:
P(x) = exp(1)*(1 - 1/4*x - 283/288*x^2 - 6223/1152*x^3 - 140734037/4147200*x^4 - 3826874463/16588800*x^5 - 290665690804549/175575859200*x^6 +...);
Q(x) = exp(1)*(1 - 5/4*x - 787/288*x^2 - 13731/1152*x^3 - 271921637/4147200*x^4 - 6765586315/16588800*x^5 - 481505562390493/175575859200*x^6 +...).

Prog

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

Crossrefs

Cf. A005169, A192728, A192729, A192730.

Sequence in context: A047149 A200757 A000260 * A125279 A376233 A186371

Adjacent sequences: A192734 A192735 A192736 * A192738 A192739 A192740

Keyword

nonn

Author

Paul D. Hanna, Jul 08 2011

Status

approved