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%I #22 Aug 06 2023 13:58:17
%S 1,1,2,7,29,131,627,3124,16032,84162,449828,2439550,13391105,74256824,
%T 415357737,2340775363,13278009018,75753246286,434392031856,
%U 2502289328542,14473290097526,84023214062635,489424396591995,2859551104564120,16754209625090980,98415932763515679,579475837597933632,3419452319373566239,20219028961691299994
%N G.f. A(x) satisfies A(x)^3 = 1 + x*A(x) + x*A(x)^2 + x*A(x)^6.
%C Compare to: G(x)^3 = 1 + x*G(x) + x*G(x)^2 + x*G(x)^3 holds when G(x) = 1/(1-x).
%H Paul D. Hanna, <a href="/A300048/b300048.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) ~ sqrt(1 + sqrt((9 + 40*sqrt(3))/13)) * (9 + 6*sqrt(3) + sqrt(153 + 100*sqrt(3)))^n / (sqrt(Pi) * n^(3/2) * 2^(n + 3/2) * 3^(n + 3/4)). - _Vaclav Kotesovec_, Aug 11 2021
%F a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(n+2*k,n-1-k) for n > 0. - _Seiichi Manyama_, Aug 05 2023
%e G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 29*x^4 + 131*x^5 + 627*x^6 + 3124*x^7 + 16032*x^8 + 84162*x^9 + 449828*x^10 + 2439550*x^11 + 13391105*x^12 + ...
%e RELATED SERIES.
%e A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 76*x^4 + 348*x^5 + 1681*x^6 + 8432*x^7 + 43495*x^8 + 229260*x^9 + 1229371*x^10 + ...
%e A(x)^6 = 1 + 6*x + 27*x^2 + 122*x^3 + 579*x^4 + 2862*x^5 + 14588*x^6 + 76146*x^7 + 405039*x^8 + 2187756*x^9 + 11967426*x^10 + ...
%e A(x)^3 = 1 + 3*x + 9*x^2 + 34*x^3 + 147*x^4 + 684*x^5 + 3341*x^6 + 16896*x^7 + 87702*x^8 + 464566*x^9 + 2501178*x^10 + ...
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = (1 + x*A + x*A^2 + x*A^6 +x*O(x^n))^(1/3) ); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A000108, A106228, A219537.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 25 2018
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