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%I #14 Apr 17 2016 06:06:30
%S 1,1,6,46,391,3519,32844,314364,3065049,30309929,303099290,3058547381,
%T 31095231708,318128139796,3272175152355,33812476576290,
%U 350804444501589,3652493334187197,38148263715573364,399552867370295155,4195305107766973240,44150591852677070280,465588059585378099226,4919039064854516328821,52059830109088065802395,551834199223958450647359,5857932269440676202573084
%N G.f. A(x) satisfies: A(x) = A( x^11 + 11*x*A(x)^11 )^(1/11), with A(0)=0, A'(0)=1.
%C Compare the g.f. to the following identities:
%C (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
%C (2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
%C where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%C 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 Paul D. Hanna, <a href="/A271933/b271933.txt">Table of n, a(n) for n = 1..300</a>
%e G.f.: A(x) = x + x^2 + 6*x^3 + 46*x^4 + 391*x^5 + 3519*x^6 + 32844*x^7 + 314364*x^8 + 3065049*x^9 + 30309929*x^10 + 303099290*x^11 + 3058547381*x^12 +...
%e where A(x)^11 = A( x^11 + 11*x*A(x)^11 ).
%e RELATED SERIES.
%e A(x)^11 = x^11 + 11*x^12 + 121*x^13 + 1331*x^14 + 14641*x^15 + 161051*x^16 + 1771561*x^17 + 19487171*x^18 + 214358881*x^19 + 2357947691*x^20 + 25937424601*x^21 + 285311670612*x^22 + 3138428376754*x^23 + 34522712144657*x^24 +...
%o (PARI) {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^11 + 11*X*A^11)^(1/11) ); polcoeff(A,n)}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A271931, A271932.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Apr 16 2016
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