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A365774 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2.

%I A365774 #19 Oct 12 2023 10:27:26
%S A365774 1,1,9,73,625,5681,53945,528697,5307489,54298849,564079337,5934390441,
%T A365774 63098046929,676976915473,7319925023897,79684985945753,
%U A365774 872620958369473,9606337027601345,106249046704511945,1180096759408431881,13156993620315230001,147193406523115480049
%N A365774 Expansion of g.f. A(x) satisfying A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2.
%C A365774 Related identities which hold formally for all Maclaurin series F(x):
%C A365774 (1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),
%C A365774 (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 A365774 (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 A365774 (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 A365774 (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 A365774 Paul D. Hanna, <a href="/A365774/b365774.txt">Table of n, a(n) for n = 0..400</a>
%F A365774 a(n) = Sum{k=0..n} binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 4^k.
%F A365774 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) * 4^k.
%F A365774 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A365774 (1) A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2.
%F A365774 (2) A(x) = (1/x) * Series_Reversion( x/(1 + x/(1 - 4*x)^2) ).
%F A365774 (3) A( x/(1 + x/(1 - 4*x)^2) ) = 1 + x/(1 - 4*x)^2.
%F A365774 (4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x)^n / (1 + (n+m-4)*x*A(x))^(n+1) for all fixed nonnegative m.
%F A365774 (4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x)^n / (1 + (n-4)*x*A(x))^(n+1).
%F A365774 (4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n-3)*x*A(x))^(n+1).
%F A365774 (4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n-2)*x*A(x))^(n+1).
%F A365774 (4.d) A(x) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x)^n / (1 + (n-1)*x*A(x))^(n+1).
%F A365774 (4.e) A(x) = 1 + 5 * Sum{n>=1} n*(n+4)^(n-2) * x^n * A(x)^n / (1 + n*x*A(x))^(n+1).
%e A365774 G.f.: A(x) = 1 + x + 9*x^2 + 73*x^3 + 625*x^4 + 5681*x^5 + 53945*x^6 + 528697*x^7 + 5307489*x^8 + 54298849*x^9 + 564079337*x^10 + ...
%e A365774 where A(x) satisfies A(x) = 1 + x*A(x)/(1 - 4*x*A(x))^2
%e A365774 also
%e A365774 A(x) = 1 + 1^0*x*A(x)/(1 + (-3)*x*A(x))^2 + 2^1*x^2*A(x)^2/(1 + (-2)*x*A(x))^3 + 3^2*x^3*A(x)^3/(1 + (-1)*x*A(x))^4 + 4^3*x^4*A(x)^4/(1 + 0*x*A(x))^5 + 5^4*x^5*A(x)^5/(1 + 1*x*A(x))^6 + 6^5*x^6*A(x)^6/(1 + 2*x*A(x))^7 + ...
%e A365774 and
%e A365774 A(x) = 1 + 5*1*5^(-1)*x*A(x)/(1 + 1*x*A(x))^2 + 5*2*6^0*x^2*A(x)^2/(1 + 2*x*A(x))^3 + 5*3*7^1*x^3*A(x)^3/(1 + 3*x*A(x))^4 + 5*4*8^2*x^4*A(x)^4/(1 + 4*x*A(x))^5 + 5*5*9^3*x^5*A(x)^5/(1 + 5*x*A(x))^6 + ...
%o A365774 (PARI) {a(n) = sum(k=0, n, binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k) * 4^k)}
%o A365774 for(n=0,30, print1(a(n),", "))
%o A365774 (PARI) {a(n) = my(A = (1/x) * serreverse( x/(1 + x/(1 - 4*x +O(x^(n+2)) )^2) ) ); polcoeff(A,n)}
%o A365774 for(n=0,30, print1(a(n),", "))
%Y A365774 Cf. A365770, A109081, A365772, A365773, A365775.
%Y A365774 Cf. A366234 (dual).
%K A365774 nonn
%O A365774 0,3
%A A365774 _Paul D. Hanna_, Oct 04 2023