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A365773 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2.

%I A365773 #21 Nov 16 2023 10:28:44
%S A365773 1,1,7,46,325,2446,19234,156115,1298077,11000584,94668508,825087418,
%T A365773 7267943962,64602794647,578726742481,5219620390558,47357456920165,
%U A365773 431941341136552,3958215409319608,36425213089790932,336475535026075180,3118885520601252016,29000562051786329512
%N A365773 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2.
%C A365773 Related identities which hold formally for all Maclaurin series F(x):
%C A365773 (1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
%C A365773 (2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),
%C A365773 (3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),
%C A365773 (4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),
%C A365773 (5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.
%H A365773 Paul D. Hanna, <a href="/A365773/b365773.txt">Table of n, a(n) for n = 0..400</a>
%F A365773 a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 3^k.
%F A365773 Let A(x)^m = Sum_{n>=0} a(n,m) * x^n then a(n,m) = Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * binomial(2*n-k-1, k) * 3^k.
%F A365773 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A365773 (1) A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2.
%F A365773 (2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 3*x)^2) ).
%F A365773 (3) A( x/(1 + x/(1 - 3*x)^2) ) = 1 + x/(1 - 3*x)^2.
%F A365773 (4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-3)*x*A(x))^(n+1) for all fixed nonnegative m.
%F A365773 (4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-3)*x*A(x))^(n+1).
%F A365773 (4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
%F A365773 (4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
%F A365773 (4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).
%F A365773 (4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n * A(x)^n / (1 + (n+1)*x*A(x))^(n+1).
%F A365773 a(n) ~ 3^(1 + 3*n) * 11^(3/2 + n) / (2*sqrt((65 - 288/(1031 + 121*sqrt(73))^(1/3) + 16*(1031 + 121*sqrt(73))^(1/3)) * Pi) * n^(3/2) * (52 - (5182*2^(2/3)) / (-174721 + 65043*sqrt(73))^(1/3) + (2*(-174721 + 65043*sqrt(73)))^(1/3))^(n + 1/2)). - _Vaclav Kotesovec_, Nov 16 2023
%e A365773 G.f.: A(x) = 1 + x + 7*x^2 + 46*x^3 + 325*x^4 + 2446*x^5 + 19234*x^6 + 156115*x^7 + 1298077*x^8 + 11000584*x^9 + 94668508*x^10 + ...
%e A365773 where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 3*x*A(x))^2
%e A365773 also
%e A365773 A(x) = 1 + 1^0*x^1*A(x)^1/(1 + (-2)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-1)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + 0*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + 1*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 2*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 3*x*A(x))^7 + ...
%e A365773 and
%e A365773 A(x) = 1 + 4*1*4^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 4*2*5^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 4*3*6^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 4*4*7^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 4*5*8^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...
%o A365773 (PARI) {a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 3^k)}
%o A365773 for(n=0,30, print1(a(n),", "))
%o A365773 (PARI) {a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 3*x +O(x^(n+2)) )^2) ) ); polcoeff(A,n)}
%o A365773 for(n=0,30, print1(a(n),", "))
%Y A365773 Cf. A365770, A109081, A365772, A365774, A365775.
%Y A365773 Cf. A366233 (dual).
%K A365773 nonn
%O A365773 0,3
%A A365773 _Paul D. Hanna_, Oct 04 2023