|
%I #11 Apr 17 2016 06:06:08
%S 1,1,3,11,44,185,803,3564,16082,73502,339391,1580318,7410356,34956846,
%T 165756814,789543189,3775883483,18122280953,87257629998,421366007784,
%U 2040186607333,9902368905093,48170863713973,234819266573684,1146894750998644,5611743950271715,27504683191546135,135022232452511063,663824940592999965,3268249153576985903,16112282609951232426,79533340761082180995
%N G.f. A(x) satisfies: A(x) = A( x^5 + 5*x*A(x)^5 )^(1/5), 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="/A271931/b271931.txt">Table of n, a(n) for n = 1..300</a>
%e G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 803*x^7 + 3564*x^8 + 16082*x^9 + 73502*x^10 + 339391*x^11 + 1580318*x^12 +...
%e where A(x)^5 = A( x^5 + 5*x*A(x)^5 ).
%e RELATED SERIES.
%e A(x)^5 = x^5 + 5*x^6 + 25*x^7 + 125*x^8 + 625*x^9 + 3126*x^10 + 15640*x^11 + 78275*x^12 + 391875*x^13 + 1962500*x^14 + 9831253*x^15 + 49265695*x^16 + 246954125*x^17 + 1238292500*x^18 + 6211046875*x^19 + 31163071886*x^20 +...
%o (PARI) {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^5 + 5*X*A^5)^(1/5) ); polcoeff(A,n)}
%o for(n=1,40,print1(a(n),", "))
%Y Cf. A271932, A271933.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Apr 16 2016
|
