cp's OEIS Frontend

A375453 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^3 ).

%I A375453 #19 Sep 11 2024 16:20:11
%S A375453 1,3,9,31,117,459,1835,7449,30711,128601,546537,2354139,10260492,
%T A375453 45173868,200578692,896865572,4033380894,18224524458,82664886074,
%U A375453 376161628302,1716301466139,7848924260901,35966629306221,165109474283847,759210907786198,3496438156668822,16126158739138860
%N A375453 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^3 ).
%C A375453 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 A375453 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 A375453 Paul D. Hanna, <a href="/A375453/b375453.txt">Table of n, a(n) for n = 1..520</a>
%F A375453 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A375453 (1) A(x)^2 = A( x^2/(1-2*x)^3 ).
%F A375453 (2) A(x)^4 = A( x^4*(1-2*x)^3 / ((1-4*x)^3*(1 - 2*x + 2*x^2)^3) ).
%F A375453 (3) A(x^2 + 2*x^3) = A( x/(1+2*x) )^2.
%F A375453 The radius of convergence r satisfies r = (1 - 2*r)^3, where A(r) = 1 and r = (1/12)*(6 + (6*sqrt(87) - 54)^(1/3) - (6*sqrt(87) + 54)^(1/3)) = 0.20512274384927080786...
%e A375453 G.f.: A(x) = x + 3*x^2 + 9*x^3 + 31*x^4 + 117*x^5 + 459*x^6 + 1835*x^7 + 7449*x^8 + 30711*x^9 + 128601*x^10 + ...
%e A375453 where A(x)^2 = A( x^2/(1-2*x)^3 ).
%e A375453 RELATED SERIES.
%e A375453 A(x)^2 = x^2 + 6*x^3 + 27*x^4 + 116*x^5 + 501*x^6 + 2178*x^7 + 9491*x^8 + 41424*x^9 + 181293*x^10 + ...
%e A375453 (A(x)/x)^(1/3) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 77*x^5 + 290*x^6 + 1122*x^7 + 4462*x^8 + 18210*x^9 + ... + A375443(n)*x^n + ...
%e A375453 x/Series_Reversion( A( x^2/(1-2*x) )^(1/2) ) = 1 + x + 2*x^2 - 2*x^4 + 6*x^6 - 20*x^8 + 70*x^10 - 263*x^12 + 1044*x^14 - 4263*x^16 + 17762*x^18 + ...
%e A375453 x/Series_Reversion( A( x^3/(1-2*x)^3 )^(1/3) ) = 1 + 2*x + x^3 - x^6 + 3*x^9 - 10*x^12 + 34*x^15 - 124*x^18 + 482*x^21 - 1931*x^24 + 7893*x^27 + ...
%e A375453 SPECIFIC VALUES.
%e A375453 A(t) = 3/4 at t = 0.201772636312778304679687617697508690090653188...
%e A375453 A(t) = 3/5 at t = 0.194614960496736155296642077884228463225576089...
%e A375453 A(t) = 1/2 at t = 0.186135869221980538627401571340819246192140850...
%e A375453 A(t) = 2/5 at t = 0.173143830263370608074654087902797631449309857...
%e A375453 A(t) = 1/4 at t = 0.140069990039210460387276300843591158073987855...
%e A375453 A(1/5) = 0.700768312277362449514797370811301885385349818...
%e A375453 where A(1/5)^2 = A(5/27).
%e A375453 A(1/6) = 0.362320684925221039201199651574198595785551012...
%e A375453 where A(1/6)^2 = A(6/64).
%e A375453 A(1/7) = 0.259569089568076471080673806323871020166140312...
%e A375453 where A(1/7)^2 = A(7/125).
%e A375453 A(1/10) = 0.14404022241542053703979110789205898915122135...
%e A375453 where A(1/10)^2 = A(10/512).
%o A375453 (PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
%o A375453 A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^3 ) - Ax^2, #A) ); A[n+1]}
%o A375453 for(n=1, 30, print1(a(n), ", "))
%Y A375453 Cf. A001006, A000108, A375454, A375455, A375456.
%Y A375453 Cf. A375443.
%K A375453 nonn
%O A375453 1,2
%A A375453 _Paul D. Hanna_, Aug 16 2024