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%I #6 Nov 30 2014 22:05:25
%S 1,1,1,1,4,10,19,49,154,415,1066,3322,10639,31279,96751,326908,
%T 1082065,3514285,12068659,42713311,148480885,523469791,1915394458,
%U 7043990266,25840482346,96913386589,370304345755,1415830556098,5458805003308,21434092603255,84865526136793,337266100883830
%N G.f. A(x) satisfies A(x) = 1 + x/(1-9*x^3)^(1/3) * A( x/sqrt(1-9*x^3)^(1/3) ).
%F G.f.: Sum_{n>=0} x^n / Product_{k=0..n} (1 - 9*k*x^3)^(1/3).
%F G.f. A(x) satisfies: A( x/sqrt(1+9*x^3)^(1/3) ) = 1 + x*A(x).
%F a(n) = Sum_{k=0..(n-1)/3} a(n-3*k-1) * binomial(n/3-1,k) * 9^k, for n>0 with a(0)=1. [After _Vladimir Kruchinin_ in A201169.]
%e G.f.: A(x) = 1 + x + x^2 + x^3 + 4*x^4 + 10*x^5 + 19*x^6 + 49*x^7 +...
%e such that A(x) = 1 + G(x)*A(G(x)) where
%e G(x) = x/(1-9*x^3)^(1/3) = x + 3*x^4 + 18*x^7 + 126*x^10 + 945*x^13 + 7371*x^16 + 58968*x^19 + 480168*x^22 +...+ A004987(n)*x^(3*n+1) +...
%e The g.f. also equals the series
%e A(x) = 1 + x/(1-9*x^3)^(1/3) + x^2/((1-9*x^3)*(1-18*x^3))^(1/3) + x^3/((1-9*x^3)*(1-18*x^3)*(1-27*x^3))^(1/3) + x^4/((1-9*x^3)*(1-18*x^3)*(1-27*x^3)*(1-36*x^3))^(1/3) +...
%o (PARI) {a(n)=polcoeff( sum(m=0, n, x^m / prod(k=0, m, (1 - 9*k*x^3 +x*O(x^n))^(1/3) )), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=if(n==0,1,sum(k=0,(n-1)\3, a(n-3*k-1) * binomial(n/3-1,k) * 3^(2*k)))}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A201169, A004987.
%K nonn
%O 0,5
%A _Paul D. Hanna_, Nov 30 2014
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