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A211894 G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^2 * x^n/n ), where Jacobsthal(n) = A001045(n). 4

%I #12 Oct 18 2020 04:55:14

%S 1,3,6,18,57,195,684,2460,8970,33102,123204,461868,1741410,6597750,

%T 25099584,95822928,366943881,1408947675,5422742910,20915079258,

%U 80820382425,312839889219,1212812010804,4708415402772,18302630040504,71230126892088,277514015733168

%N G.f.: exp( Sum_{n>=1} 3 * Jacobsthal(n)^2 * 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)^(2/3) / ((1-x)*(1-4*x))^(1/3).

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

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

%e G.f.: A(x) = 1 + 3*x + 6*x^2 + 18*x^3 + 57*x^4 + 195*x^5 + 684*x^6 +...

%e such that

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

%e Jacobsthal numbers begin:

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

%t CoefficientList[Series[(1+2*x)^(2/3) / ((1-x)*(1-4*x))^(1/3), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Oct 18 2020 *)

%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)^2*x^k/k)+x*O(x^n)), n)}

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

%o (PARI) {a(n)=polcoeff(((1+2*x)^2/((1-x)*(1-4*x) +x*O(x^n)))^(1/3),n)}

%Y Cf. A211893, A211895, A211896, A054888, A207969, A001045 (Jacobsthal).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 25 2012

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