A211896 - OEIS

%I #14 Oct 18 2020 05:02:43

%S 1,3,6,90,723,10689,130428,1862580,25594611,368313993,5289203262,

%T 77279744418,1134460916361,16798605635235,249994099311288,

%U 3740771822960664,56208829313956998,847934859174601650,12834366187138678836,194855374723972622988,2966358133685609559042

%N G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^4 * x^n/n ), where Jacobsthal(n) = A001045(n).

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

%F G.f.: ( (1+2*x)^4*(1+8*x)^4 / ((1-x)*(1-4*x)^6*(1-16*x)) )^(1/27).

%F G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^4 / 27 * x^n/n ).

%F a(n) ~ 3^(5/27) * 2^(4*n) / (5^(1/27) * Gamma(1/27) * n^(26/27)). - _Vaclav Kotesovec_, Oct 18 2020

%e G.f.: A(x) = 1 + 3*x + 6*x^2 + 90*x^3 + 723*x^4 + 10689*x^5 + 130428*x^6 +...

%e such that

%e log(A(x))/3 = x + x^2/2 + 3^4*x^3/3 + 5^4*x^4/4 + 11^4*x^5/5 + 21^4*x^6/6 + 43^4*x^7/7 +...+ Jacobsthal(n)^4*x^n/n +...

%e Jacobsthal numbers begin:

%e A001045 = [1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,...].

%o (PARI) {Jacobsthal(n)=polcoeff(x/(1-x-2*x^2+x*O(x^n)),n)}

%o {a(n)=polcoeff(exp(sum(k=1, n, 3*Jacobsthal(k)^4*x^k/k)+x*O(x^n)), n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=polcoeff(((1+2*x)^4*(1+8*x)^4/((1-x)*(1-4*x)^6*(1-16*x))+x*O(x^n))^(1/27),n)}

%Y Cf. A211893, A211894, A211895, A207969, A001045 (Jacobsthal).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 25 2012