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A367724 Expansion of g.f. A(x) satisfying x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^4) such that A(0) = 1.

%I A367724 #13 Mar 28 2025 06:07:45
%S A367724 1,1,4,18,90,487,2785,16559,101309,633523,4030964,26012632,169842909,
%T A367724 1119960451,7447798584,49891581704,336355963821,2280413834539,
%U A367724 15538054908152,106345134539952,730770552189188,5039872364942277,34872958276988909,242026790074533733,1684356151039022090
%N A367724 Expansion of g.f. A(x) satisfying x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^4) such that A(0) = 1.
%F A367724 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A367724 (1) x = A(x) * (1 - A(x)) / (1 - A(x) - A(x)^4).
%F A367724 (2) x = (1+x)*A(x) - A(x)^2 + x*A(x)^4 such that A(0) = 1.
%F A367724 (3) A(x) = x / Series_Reversion(x*(1 + Series_Reversion( x/((1 + x)^4 + x) ))).
%F A367724 (4) a(n) = (Sum_{k=1..n} binomial(n, k) * binomial(4*k-n, k-1))/n for n > 0 with a(0) = 1 (derived from a formula by _Tani Akinari_ in A243156).
%e A367724 G.f. A(x) = 1 + x + 4*x^2 + 18*x^3 + 90*x^4 + 487*x^5 + 2785*x^6 + 16559*x^7 + 101309*x^8 + 633523*x^9 + 4030964*x^10 + ...
%e A367724 Let R(x) = x * (1 - x) / (1 - x - x^4) then R(A(x)) = x;
%e A367724 however, A(R(x)) does not equal x, rather
%e A367724 A(R(x)) = 1 + x + 4*x^2 + 18*x^3 + 90*x^4 + 488*x^5 + 2794*x^6 + 16622*x^7 + 101732*x^8 + 636382*x^9 + 4050546*x^10 + ...
%o A367724 (PARI) {a(n)=polcoeff(x/serreverse(x*(1+serreverse(x/((1 + x)^4 + x +x*O(x^n))))), n)}
%o A367724 for(n=0, 30, print1(a(n), ", "))
%o A367724 (PARI) /* From a formula by Tani Akinari in A243156 */
%o A367724 {a(n)=0^n+sum(k=1, n, binomial(n, k)*binomial(4*k-n, k-1))/(n+0^n)}
%o A367724 for(n=0,30,print1(a(n),", "))
%Y A367724 Cf. A243156, A367725.
%K A367724 nonn
%O A367724 0,3
%A A367724 _Paul D. Hanna_, Nov 28 2023