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A271932 G.f. A(x) satisfies: A(x) = A( x^7 + 7*x*A(x)^7 )^(1/7), with A(0)=0, A'(0)=1.

%I A271932 #14 Apr 17 2016 06:05:41
%S A271932 1,1,4,20,110,638,3828,23515,146970,930820,5957325,38452405,249944939,
%T A271932 1634287025,10739831400,70884562683,469622328252,3121694320866,
%U A271932 20811920304961,139115729296575,932107335003790,6258662787526655,42105353650697301,283765005631661148,1915495724241980280,12949332513585521217,87661142189041380207,594176943178375193748,4032121696383579351905,27392082325012470506385,186276500908841717917320
%N A271932 G.f. A(x) satisfies: A(x) = A( x^7 + 7*x*A(x)^7 )^(1/7), with A(0)=0, A'(0)=1.
%C A271932 Compare the g.f. to the following identities:
%C A271932 (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
%C A271932 (2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
%C A271932 where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%C A271932 More generally, for prime p there exists an integer series G(x) that satisfies: G(x) = G( x^p + p*x*G(x)^p )^(1/p) with G(0)=0, G'(0)=1 (conjecture).
%H A271932 Paul D. Hanna, <a href="/A271932/b271932.txt">Table of n, a(n) for n = 1..300</a>
%e A271932 G.f.: A(x) = x + x^2 + 4*x^3 + 20*x^4 + 110*x^5 + 638*x^6 + 3828*x^7 + 23515*x^8 + 146970*x^9 + 930820*x^10 + 5957325*x^11 + 38452405*x^12 +...
%e A271932 where A(x)^7 = A( x^7 + 7*x*A(x)^7 ).
%e A271932 RELATED SERIES.
%e A271932 A(x)^7 = x^7 + 7*x^8 + 49*x^9 + 343*x^10 + 2401*x^11 + 16807*x^12 + 117649*x^13 + 823544*x^14 + 5764822*x^15 + 40353901*x^16 + 282478679*x^17 + 1977362758*x^18 + 13841640148*x^19 + 96892304579*x^20 + 678252720401*x^21 +...
%o A271932 (PARI) {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^7 + 7*X*A^7)^(1/7) ); polcoeff(A,n)}
%o A271932 for(n=1,40,print1(a(n),", "))
%Y A271932 Cf. A271931, A271933.
%K A271932 nonn
%O A271932 1,3
%A A271932 _Paul D. Hanna_, Apr 16 2016