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
Keywords
Examples
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,...].
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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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 *) -
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)}
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
Comments