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A375454 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^4 ).

%I A375454 #12 Sep 11 2024 16:20:21
%S A375454 1,4,14,56,253,1188,5598,26456,126278,611640,3010948,15058064,
%T A375454 76399263,392524428,2038142346,10674131464,56281299098,298286598920,
%U A375454 1586959692508,8466559886448,45260129274602,242296862848648,1298487003814300,6964374684442416,37378818578434617,200745991803248388
%N A375454 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^4 ).
%C A375454 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 A375454 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 A375454 Paul D. Hanna, <a href="/A375454/b375454.txt">Table of n, a(n) for n = 1..400</a>
%F A375454 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A375454 (1) A(x)^2 = A( x^2/(1-2*x)^4 ).
%F A375454 (2) A(x)^4 = A( x^4*(1-2*x)^8/((1-2*x)^4 - 2*x^2)^4 ).
%F A375454 (3) A(x^2 + 4*x^3 + 4*x^4) = A( x/(1+2*x) )^2.
%F A375454 The radius of convergence r satisfies r = (1 - 2*r)^4, where A(r) = 1 and r = 0.17610056436947880725475...
%e A375454 G.f.: A(x) = x + 4*x^2 + 14*x^3 + 56*x^4 + 253*x^5 + 1188*x^6 + 5598*x^7 + 26456*x^8 + 126278*x^9 + 611640*x^10 + ...
%e A375454 where A(x)^2 = A( x^2/(1-2*x)^4 ).
%e A375454 RELATED SERIES.
%e A375454 A(x)^2 = x^2 + 8*x^3 + 44*x^4 + 224*x^5 + 1150*x^6 + 5968*x^7 + 30920*x^8 + 159296*x^9 + 818013*x^10 + ...
%e A375454 (A(x)/x)^(1/2) = 1 + 2*x + 5*x^2 + 18*x^3 + 78*x^4 + 348*x^5 + 1551*x^6 + 6982*x^7 + 32114*x^8 + 151620*x^9 + 734458*x^10 + ...
%e A375454 (A(x)/x)^(1/4) = 1 + x + 2*x^2 + 7*x^3 + 30*x^4 + 130*x^5 + 561*x^6 + 2460*x^7 + 11115*x^8 + 51948*x^9 + 250551*x^10 + ...
%e A375454 x/Series_Reversion( A( x^4/(1-2*x)^4 )^(1/4) ) = 1 + 2*x + x^4 - 2*x^8 + 9*x^12 - 46*x^16 + 251*x^20 - 1467*x^24 + 9001*x^28 - 56961*x^32 + 369035*x^36 + ...
%e A375454 SPECIFIC VALUES.
%e A375454 A(t) = 3/4 at t = 0.1736940609090204697398931237543698538457793088071...
%e A375454 A(t) = 3/5 at t = 0.1683900940132911881249251740473132322447213579710...
%e A375454 A(t) = 1/2 at t = 0.1619792168710406277922262531667140037108384145069...
%e A375454 A(t) = 2/5 at t = 0.1519644942152899698264943158671103683281536948439...
%e A375454 A(t) = 1/4 at t = 0.1256102247935771858127425480259396391143977190820...
%e A375454 A(1/6) = 0.56831064552606196969057398988138945640151282529741...
%e A375454 where A(1/6)^2 = A(9/64).
%e A375454 A(1/7) = 0.33620864108171743638518920200481354359529601205566...
%e A375454 where A(1/7)^2 = A(49/625).
%e A375454 A(1/8) = 0.24749365203461325611367946855098276802746512221191...
%e A375454 where A(1/8)^2 = A(4/81).
%e A375454 A(1/9) = 0.19726025709005021216217281111162473370492543981591...
%e A375454 where A(1/9)^2 = A(81/2401).
%e A375454 A(1/10) = 0.1643908422523431149995416239752018754267879073230...
%e A375454 where A(1/10)^2 = A(25/1024).
%o A375454 (PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
%o A375454 A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^4 ) - Ax^2, #A) ); A[n+1]}
%o A375454 for(n=1, 30, print1(a(n), ", "))
%Y A375454 Cf. A001006, A000108, A375453, A375455, A375456.
%K A375454 nonn
%O A375454 1,2
%A A375454 _Paul D. Hanna_, Aug 17 2024