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A271935 G.f. A(x) satisfies: A(x) = A( x^2 + 8*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

%I A271935 #13 May 21 2024 05:30:28
%S A271935 1,4,26,200,1691,15204,142710,1382568,13721765,138802136,1425785270,
%T A271935 14832383488,155947271878,1654494195340,17690004381000,
%U A271935 190426309700616,2062071992480208,22447191471665160,245501068961175090,2696300196714320520,29725402250477117175,328835072363241763920
%N A271935 G.f. A(x) satisfies: A(x) = A( x^2 + 8*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.
%C A271935 Compare the g.f. to the following related identities:
%C A271935 (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), where C(x) = x + C(x)^2 (A000108).
%C A271935 (2) F(x) = F( x^2 + 4*x*F(x)^2 )^(1/2), where F(x) = D(x)^2/x and D(x) = x + D(x)^3/x (A001764).
%H A271935 Paul D. Hanna, <a href="/A271935/b271935.txt">Table of n, a(n) for n = 1..300</a>
%F A271935 G.f. A(x) satisfies: A( x*G(x^2) - 4*x^2 ) = x, where G(x)^2 = G(x^2) + 12*x, and G(x) is the g.f. of A264413.
%e A271935 G..f.: A(x) = x + 4*x^2 + 26*x^3 + 200*x^4 + 1691*x^5 + 15204*x^6 + 142710*x^7 + 1382568*x^8 + 13721765*x^9 + 138802136*x^10 + 1425785270*x^11 + ...
%e A271935 where A(x)^2 = A( x^2 + 8*x*A(x)^2 ).
%e A271935 RELATED SERIES.
%e A271935 A(x)^2 = x^2 + 8*x^3 + 68*x^4 + 608*x^5 + 5658*x^6 + 54336*x^7 + 534984*x^8 + 5373824*x^9 + 54866075*x^10 + 567775856*x^11 + 5942353444*x^12 + ...
%e A271935 Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
%e A271935 B(x) = x - 4*x^2 + 6*x^3 - 15*x^5 + 90*x^7 - 660*x^9 + 5310*x^11 - 45765*x^13 + 413640*x^15 - 3864345*x^17 + 37014120*x^19 + ... + A264413(n)*x^(2*n+1) + ...
%e A271935 such that B(x) = x*G(x^2) - 4*x^2 where G(x)^2 = G(x^2) + 12*x, and G(x) is the g.f. of A264413.
%e A271935 From _Paul D. Hanna_, May 20 2024: (Start)
%e A271935 The series (A(x)/x)^(1/4) seems to consist solely of integer coefficients
%e A271935 (A(x)/x)^(1/4) = 1 + x + 5*x^2 + 34*x^3 + 268*x^4 + 2305*x^5 + 20988*x^6 + 198891*x^7 + 1941111*x^8 + 19377707*x^9 + 196936775*x^10 + ...
%e A271935 and continues to be integral for at least the initial 400 coefficients. (End)
%o A271935 (PARI) {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 8*X*A^2)^(1/2) ); polcoeff(A,n)}
%o A271935 for(n=1,30,print1(a(n),", "))
%Y A271935 Cf. A264413, A271930, A271957, A271931, A271934.
%K A271935 nonn
%O A271935 1,2
%A A271935 _Paul D. Hanna_, Apr 16 2016