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A375455 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 ).

%I A375455 #10 Sep 11 2024 16:20:32
%S A375455 1,5,20,90,470,2566,13885,74435,400530,2183930,12112167,68351005,
%T A375455 392055575,2281947435,13450584580,80110426698,481032299830,
%U A375455 2905955107950,17629836770715,107254895106265,653597751574541,3986386422481665,24321398369358070,148386468804372420,905156432977350225
%N A375455 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 ).
%C A375455 Compare to: F(x)^2 = F( x^2/(1-2*x) ), where F(x) = x*M(x) and M(x) = 1 + x*M(x) + x^2*M(x)^2 is the Motzkin function (A001006).
%C A375455 Compare to: G(x)^2 = G( x^2/(1-2*x)^2 ), where G(x) = x*C(x)^2 and C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
%H A375455 Paul D. Hanna, <a href="/A375455/b375455.txt">Table of n, a(n) for n = 1..400</a>
%F A375455 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A375455 (1) A(x)^2 = A( x^2/(1-2*x)^5 ).
%F A375455 (2) A(x)^4 = A( x^4*(1-2*x)^15 / ((1-2*x)^5 - 2*x^2)^5 ).
%F A375455 (3) A(x^2*(1 + 2*x)^3) = A( x/(1+2*x) )^2.
%F A375455 The radius of convergence r satisfies r = (1 - 2*r)^5, where A(r) = 1 and r = 0.155430268881057887439965...
%e A375455 G.f.: A(x) = x + 5*x^2 + 20*x^3 + 90*x^4 + 470*x^5 + 2566*x^6 + 13885*x^7 + 74435*x^8 + 400530*x^9 + 2183930*x^10 + ...
%e A375455 where A(x)^2 = A( x^2/(1-2*x)^5 ).
%e A375455 RELATED SERIES.
%e A375455 A(x)^2 = x^2 + 10*x^3 + 65*x^4 + 380*x^5 + 2240*x^6 + 13432*x^7 + 80330*x^8 + 474960*x^9 + 2783590*x^10 + ...
%e A375455 (A(x)/x)^(1/5) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 205*x^5 + 989*x^6 + 4785*x^7 + 23881*x^8 + 124245*x^9 + 673020*x^10 + ...
%e A375455 x/Series_Reversion( A( x^5/(1-2*x)^5 )^(1/5) ) = 1 + 2*x + x^5 - 3*x^10 + 18*x^15 - 124*x^20 + 925*x^25 - 7372*x^30 + 61466*x^35 - 528678*x^40 + 4656736*x^45 + ...
%e A375455 SPECIFIC VALUES.
%e A375455 A(t) = 3/4 at t = 0.1535987460670222421700476984635848015956844093413...
%e A375455 A(t) = 3/5 at t = 0.1494534252284609931621062683479802340037678508370...
%e A375455 A(t) = 1/2 at t = 0.1443598468225794843508026942502138500132562159005...
%e A375455 A(t) = 2/5 at t = 0.1362812665991487577089709044456104123756230678872...
%e A375455 A(t) = 1/4 at t = 0.1144692674833411472616636812900607840273720167873...
%e A375455 A(1/7) = 0.477612316813393143429515106540189409592882329142...
%e A375455 where A(1/7)^2 = A(343/3125).
%e A375455 A(1/8) = 0.309560069127977498956512592550239740786137843207...
%e A375455 where A(1/8)^2 = A(16/243).
%e A375455 A(1/9) = 0.234151149075763124751821214511435118422621268792...
%e A375455 where A(1/9)^2 = A(729/16807).
%e A375455 A(1/10) = 0.189302960006249918030251616127177047165765112599...
%e A375455 where A(1/10)^2 = A(125/4096).
%o A375455 (PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
%o A375455 A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^5 ) - Ax^2, #A) ); H=Ax; A[n+1]}
%o A375455 for(n=1, 30, print1(a(n), ", "))
%Y A375455 Cf. A001006, A000108, A375453, A375454, A375456.
%K A375455 nonn
%O A375455 1,2
%A A375455 _Paul D. Hanna_, Aug 17 2024