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%I #17 May 21 2024 05:30:36
%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
%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..500</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 + ...
%e Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then R(x) begins
%e R(x) = x - x^2 - x^3 - x^4 + 4*x^6 + 15*x^7 + 36*x^8 + 55*x^9 - 359*x^11 - 1520*x^12 - 4028*x^13 - 6667*x^14 + 49062*x^16 + 217645*x^17 + ...
%e The 4 quintisections of R(x) = Q1(x) + Q2(x) + Q3(x) + Q4(x) (with the fifth being zero) are as follows
%e Q1(x) = x + 4*x^6 - 359*x^11 + 49062*x^16 - 8013396*x^21 + 1442958557*x^26 - 276352605126*x^31 + 55224710824185*x^36 - 11384289478228711*x^41 + ...
%e Q2(x) = -x^2 + 15*x^7 - 1520*x^12 + 217645*x^17 - 36405005*x^22 + 6650838668*x^27 - 1286179025729*x^32 + 258819346825534*x^37 + ...
%e Q3(x) = -x^3 + 36*x^8 - 4028*x^13 + 600254*x^18 - 102567034*x^23 + 18988120493*x^28 - 3705388523045*x^33 + 750546817970646*x^38 + ...
%e Q4(x) = -x^4 + 55*x^9 - 6667*x^14 + 1028514*x^19 - 179152944*x^24 + 33573744984*x^29 - 6607215559460*x^34 + 1346634048063165*x^39 + ...
%e where Q1*Q4 = -Q2*Q3 where
%e Q2*Q3 = x^5 - 51*x^10 + 6088*x^15 - 933039*x^20 + 161933629*x^25 - 30277104991*x^30 + 5949003081867*x^35 - 1211076410858363*x^40 + ...
%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
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