A302100 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n * A(x)^n * Product_{k=0..n-1} (3*k + 1).

1, 1, 5, 41, 449, 6081, 98133, 1846377, 39888353, 977117825, 26839621829, 818332799593, 27443807417569, 1004188344449473, 39809506543659477, 1699473112658002089, 77716022374143303489, 3789578550994707778305, 196255782523222432943109, 10756748528551996006448553, 622036345094017435642828161, 37846075344692579622469742529

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. A(x) satisfies:

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

(2) A(x) = (1/x)*Series_Reversion( x/F(x) ), where F(x) = Sum_{n>=0} A007559(n)*x^n, the o.g.f. of the triple factorials.

(3) A(x) = 1 + x*A(x)^2 * (A(x) + 4*x*A'(x)) / (A(x) + x*A'(x)).

(4) A(x) = 1/(1 - x*A(x)/(1 - 3*x*A(x)/(1 - 4*x*A(x)/(1 - 6*x*A(x)/(1 - 7*x*A(x)/(1 - 9*x*A(x)/(1 - 10*x*A(x)/(1 - ...)))))))), a continued fraction.

a(n) ~ sqrt(2*Pi) * 3^n * n^(n - 1/6) / (Gamma(1/3) * exp(n - 1/3)). - Vaclav Kotesovec, Jun 18 2019

Example

G.f.: A(x) = 1 + x + 5*x^2 + 41*x^3 + 449*x^4 + 6081*x^5 + 98133*x^6 + 1846377*x^7 + 39888353*x^8 + 977117825*x^9 + 26839621829*x^10 + ...
such that
A(x) = 1 + x*A(x) + 4*x^2*A(x)^2 + 28*x^3*A(x)^3 + 280*x^4*A(x)^4 + 3640*x^5*A(x)^5 + 58240*x^6*A(x)^6 + ... + x^n * A(x)^n * Product_{k=0..n-1} (3*k + 1) + ...

Prog

(PARI) /* Series Reversion of Triple Factorials g.f.: */
{a(n) = polcoeff((1/x) * serreverse(x/sum(m=0, n, x^m * prod(k=0, m-1, 3*k + 1)) +x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Differential Equation: */
{a(n) = my(A=1); for(i=0, n, A = 1 + x*A^2*(A + 4*x*A')/(x*A +x^2*O(x^n))'); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* Continued fraction: */
{a(n) = my(A=1, CF = 1+x +x*O(x^n)); for(i=1, n, A=CF; for(k=0, n, CF = 1/(1 - floor(3*(n-k+1)/2)*x*A*CF ) )); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A007559, A088368, A301363, A302535, A302565.

Sequence in context: A083073 A115257 A225095 * A222081 A047735 A096364

Adjacent sequences: A302097 A302098 A302099 * A302101 A302102 A302103

Keyword

nonn

Author

Paul D. Hanna, Apr 09 2018

Status

approved