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A364336 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^3).

%I A364336 #23 Sep 11 2024 05:47:47
%S A364336 1,2,7,39,242,1634,11631,85957,653245,5072862,40077807,321106623,
%T A364336 2602911282,21308131235,175909559897,1462846379247,12242600576066,
%U A364336 103035285071630,871490142773640,7404121610615520,63157400073057627,540689217572662413,4644083121177225292
%N A364336 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^3).
%H A364336 Seiichi Manyama, <a href="/A364336/b364336.txt">Table of n, a(n) for n = 0..1000</a>
%F A364336 a(n) = Sum_{k=0..n} binomial(3*k+1,k) * binomial(3*k+1,n-k) / (3*k+1).
%F A364336 D-finite with recurrence -2*n*(2*n+1)*a(n) +(3*n^2+23*n-14)*a(n-1) +(207*n^2 -635*n +494)*a(n-2) +2*(397*n^2 -2031*n +2600)*a(n-3) +6*(75*n-244) *(3*n-11)*a(n-4) +9*(45*n-179) *(3*n-14)*a(n-5) +63*(3*n-14) *(3*n-17)*a(n-6) +12*(3*n-16) *(3*n-20)*a(n-7)=0. - _R. J. Mathar_, Jul 25 2023
%F A364336 From _Peter Bala_, Sep 10 2024: (Start)
%F A364336 x/series_reversion(x*A(x)) = 1 + 2*x + 3*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + ..., the g.f. of A216359.
%F A364336 (1/x) * series_reversion(x/A(x)) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + 8551*x^5 + ..., the g.f. of A215623. (End)
%p A364336 A364336 := proc(n)
%p A364336     add( binomial(3*k+1,k) * binomial(3*k+1,n-k)/(3*k+1),k=0..n) ;
%p A364336 end proc:
%p A364336 seq(A364336(n),n=0..80); # _R. J. Mathar_, Jul 25 2023
%t A364336 nmax = 80; A[_] = 1;
%t A364336 Do[A[x_] = (1 + x)*(1 + x*A[x]^3) + O[x]^(nmax+1) // Normal, {nmax+1}];
%t A364336 CoefficientList[A[x], x] (* _Jean-François Alcover_, Mar 03 2024 *)
%o A364336 (PARI) a(n) = sum(k=0, n, binomial(3*k+1, k)*binomial(3*k+1, n-k)/(3*k+1));
%Y A364336 Cf. A073157, A364337, A364338, A364339.
%Y A364336 Cf. A198953, A212071, A215623, A215654, A216359.
%K A364336 nonn
%O A364336 0,2
%A A364336 _Seiichi Manyama_, Jul 19 2023