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A365775 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2.

%I A365775 #21 Oct 12 2023 10:27:53
%S A365775 1,1,11,106,1061,11226,124026,1414211,16515981,196551736,2375042076,
%T A365775 29062573926,359407971786,4484868410231,56399986492661,
%U A365775 714067825064426,9094408567049701,116436367409647736,1497734068943432856,19346547929074098836,250851388061224003276
%N A365775 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2.
%C A365775 Related identities which hold formally for all Maclaurin series F(x):
%C A365775 (1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
%C A365775 (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 A365775 (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 A365775 (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 A365775 (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 A365775 Paul D. Hanna, <a href="/A365775/b365775.txt">Table of n, a(n) for n = 0..400</a>
%F A365775 a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 5^k.
%F A365775 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) * 5^k.
%F A365775 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A365775 (1) A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2.
%F A365775 (2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 5*x)^2) ).
%F A365775 (3) A( x/(1 + x/(1 - 5*x)^2) ) = 1 + x/(1 - 5*x)^2.
%F A365775 (4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-5)*x*A(x))^(n+1) for all fixed nonnegative m.
%F A365775 (4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-5)*x*A(x))^(n+1).
%F A365775 (4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-4)*x*A(x))^(n+1).
%F A365775 (4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n-3)*x*A(x))^(n+1).
%F A365775 (4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
%F A365775 (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 A365775 (4.f) A(x) = 1 + 6 * Sum{n>=1} n*(n+5)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).
%F A365775 a(n) ~ sqrt(3) * 5^(2*n) * (19^(3/2 + n) / (2*sqrt((113 + (28*(47225 + 1083*sqrt(1905))^(1/3))/5^(2/3) - 2632/(5*(47225 + 1083*sqrt(1905)))^(1/3))*Pi) * n^(3/2) * (68 + (2*(-1496331 + 60325*sqrt(1905)))^(1/3)/3^(2/3) - 9214*2^(2/3)/(3*(-1496331 + 60325*sqrt(1905)))^(1/3))^(n + 1/2))). - _Vaclav Kotesovec_, Oct 06 2023
%e A365775 G.f.: A(x) = 1 + x + 11*x^2 + 106*x^3 + 1061*x^4 + 11226*x^5 + 124026*x^6 + 1414211*x^7 + 16515981*x^8 + 196551736*x^9 + 2375042076*x^10 + ...
%e A365775 where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 5*x*A(x))^2
%e A365775 also
%e A365775 A(x) = 1 + 1^0*x*A(x)/(1 + (-4)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-3)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + (-2)*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 + 0*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 1*x*A(x))^7 + ...
%e A365775 and
%e A365775 A(x) = 1 + 6*1*6^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 6*2*7^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 6*3*8^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 6*4*9^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 6*5*10^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...
%o A365775 (PARI) {a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 5^k)}
%o A365775 for(n=0,30, print1(a(n),", "))
%o A365775 (PARI) {a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 5*x +O(x^(n+2)) )^2) ) ); polcoeff(A,n)}
%o A365775 for(n=0,30, print1(a(n),", "))
%Y A365775 Cf. A365770, A109081, A365772, A365773, A365774.
%Y A365775 Cf. A366235 (dual).
%K A365775 nonn
%O A365775 0,3
%A A365775 _Paul D. Hanna_, Oct 04 2023