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A370545 Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).

%I A370545 #13 Mar 27 2024 21:24:59
%S A370545 1,1,4,21,125,801,5388,37518,268109,1955000,14487754,108794169,
%T A370545 826054062,6331064385,48914088750,380555960864,2978892961194,
%U A370545 23444095375593,185394136871818,1472396312841250,11739089289817538,93921736129064325,753845680317416682,6068255413854119432
%N A370545 Expansion of g.f. A(x) satisfying A(x) = A( x^5 + 5*A(x)^6 )^(1/5).
%C A370545 Compare the g.f. to the following identities:
%C A370545 (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
%C A370545 (2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
%C A370545 where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H A370545 Paul D. Hanna, <a href="/A370545/b370545.txt">Table of n, a(n) for n = 1..400</a>
%e A370545 G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 125*x^5 + 801*x^6 + 5388*x^7 + 37518*x^8 + 268109*x^9 + 1955000*x^10 + 14487754*x^11 + 108794169*x^12 + ...
%e A370545 where A(x)^5 = A( x^5 + 5*A(x)^6 ).
%e A370545 RELATED SERIES.
%e A370545 A(x)^5 = x^5 + 5*x^6 + 30*x^7 + 195*x^8 + 1330*x^9 + 9376*x^10 + 67720*x^11 + ...
%e A370545 A(x)^6 = x^6 + 6*x^7 + 39*x^8 + 266*x^9 + 1875*x^10 + 13542*x^11 + 99654*x^12 + ...
%e A370545 Let B(x) be the series reversion of A(x), A(B(x)) = x, which begins
%e A370545 B(x) = x - x^2 - 2*x^3 - 6*x^4 - 21*x^5 - 80*x^6 - 320*x^7 - 1326*x^8 - 5637*x^9 - 24434*x^10 - ... + (-1)^(n-1)*A352703(n-1)*x^n + ...
%e A370545 then B(x)^5 + 5*x^6 = B(x^5).
%e A370545 Let C(x) = x^2/B(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 802*x^7 + 3553*x^8 + 15994*x^9 + 72886*x^10 + ... + A091200(n-1)*x^n + ...
%e A370545 where A(x^2/C(x)) = x and C(A(x)) = A(x)^2/x,
%e A370545 then C(x)^5 = C(x^5)/(1 - 5*C(x^5)/x^4).
%o A370545 (PARI) {a(n) = my(A=x+x^2); for(m=1, n, A=truncate(A); A = subst(A, x, x^5 + 5*A^6 +x^5*O(x^m))^(1/5) ); polcoeff(A, n)}
%o A370545 for(n=1, 40, print1(a(n), ", "))
%Y A370545 Cf. A352703, A091200, A271931, A370546.
%K A370545 nonn
%O A370545 1,3
%A A370545 _Paul D. Hanna_, Mar 26 2024