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 A271931 G.f. A(x) satisfies: A(x) = A( x^5 + 5*x*A(x)^5 )^(1/5), with A(0)=0, A'(0)=1. 7
 1, 1, 3, 11, 44, 185, 803, 3564, 16082, 73502, 339391, 1580318, 7410356, 34956846, 165756814, 789543189, 3775883483, 18122280953, 87257629998, 421366007784, 2040186607333, 9902368905093, 48170863713973, 234819266573684, 1146894750998644, 5611743950271715, 27504683191546135, 135022232452511063, 663824940592999965, 3268249153576985903, 16112282609951232426, 79533340761082180995 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Compare the g.f. to the following identities: (1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2), (2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3), where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108). 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). LINKS Paul D. Hanna, Table of n, a(n) for n = 1..300 EXAMPLE 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 +... where A(x)^5 = A( x^5 + 5*x*A(x)^5 ). RELATED SERIES. 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 +... PROG (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)} for(n=1, 40, print1(a(n), ", ")) CROSSREFS Cf. A271932, A271933. Sequence in context: A113174 A132840 A091200 * A151105 A127632 A061706 Adjacent sequences:  A271928 A271929 A271930 * A271932 A271933 A271934 KEYWORD nonn AUTHOR Paul D. Hanna, Apr 16 2016 STATUS approved

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)