A211895 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^3 * x^n/n ), where Jacobsthal(n) = A001045(n).

1, 3, 6, 36, 186, 1254, 8208, 57540, 404619, 2913705, 21146694, 155231256, 1147302756, 8538393900, 63879354096, 480212156664, 3624581868297, 27456690186507, 208644709097070, 1589982296208492, 12147079485362406, 93012131704072698, 713676733469348352

Offset

0,2

Comments

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f.: ( (1+x)*(1+4*x)^3 / ((1-2*x)^3*(1-8*x)) )^(1/9).

G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^3 / 9 * x^n/n ).

Recurrence: n*a(n) = (5*n-2)*a(n-1) + 6*(5*n-12)*a(n-2) - 8*(5*n-12)*a(n-3) - 64*(n-4)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ 3^(2/9)*8^n/(Gamma(1/9)*n^(8/9)). - Vaclav Kotesovec, Oct 24 2012

Example

G.f.: A(x) = 1 + 3*x + 6*x^2 + 36*x^3 + 186*x^4 + 1254*x^5 + 8208*x^6 +...
such that
log(A(x))/3 = x + x^2/2 + 3^3*x^3/3 + 5^3*x^4/4 + 11^3*x^5/5 + 21^3*x^6/6 + 43^3*x^7/7 +...+ Jacobsthal(n)^3*x^n/n +...
Jacobsthal numbers begin:
A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

Mathematica

CoefficientList[Series[((1+x)*(1+4*x)^3/((1-2*x)^3*(1-8*x)))^(1/9), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Prog

(PARI) {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)), n)}
{a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^3*x^k/k)+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(( (1+x)*(1+4*x)^3 / ((1-2*x)^3*(1-8*x)+x*O(x^n)) )^(1/9), n)}

Crossrefs

Cf. A211893, A211894, A211896, A207970, A001045 (Jacobsthal).

Sequence in context: A076983 A068084 A003674 * A240986 A372003 A120595

Adjacent sequences: A211892 A211893 A211894 * A211896 A211897 A211898

Keyword

nonn

Author

Paul D. Hanna, Apr 25 2012

Status

approved