This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A375444 #15 Aug 21 2025 14:41:13
%S A375444 1,1,2,7,30,130,561,2460,11115,51948,250551,1240828,6274580,32231322,
%T A375444 167460901,876998437,4617448333,24395086617,129162020323,684753458054,
%U A375444 3633159683023,19287528099428,102441443882448,544372928359375,2894576197980724,15402989792369740,82040643327234351
%N A375444 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^4 )/(1-2*x).
%C A375444 Compare to M(x)^2 = M( x^2/(1-2*x) )/(1-2*x), where M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).
%C A375444 Compare to C(x)^2 = C( x^2/(1-2*x)^2 )/(1-2*x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%H A375444 Paul D. Hanna, <a href="/A375444/b375444.txt">Table of n, a(n) for n = 0..400</a>
%F A375444 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A375444 (1) A(x)^2 = A( x^2/(1-2*x)^4 )/(1-2*x).
%F A375444 (2) A(x)^4 = A( x^4*y^4 )*y where y = (1-2*x)^2/((1-2*x)^4 - 2*x^2).
%F A375444 (3) A(x^2 + 4*x^3 + 4*x^4) = A( x/(1+2*x) )^2 / (1+2*x).
%F A375444 The radius of convergence r satisfies r = (1 - 2*r)^4, where A(r) = 1/(1-2*r) and r = 0.17610056436947880725475085178711534652...
%e A375444 G.f.: A(x) = 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 A375444 where A(x)^2 = A( x^2/(1-2*x)^4 )/(1-2*x).
%e A375444 RELATED SERIES.
%e A375444 A(x)^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 A375444 A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 253*x^4 + 1188*x^5 + 5598*x^6 + 26456*x^7 + 126278*x^8 + ... + A375454(n+1)*x^n + ...
%e A375444 SPECIFIC VALUES.
%e A375444 Given the radius of convergence r = 0.17610056436947880725475...,
%e A375444 A(r) = 1.5436890126920763615708559718017479865252032976509...
%e A375444 where r = (1-2*r)^4 and A(r) = 1/(1-2*r).
%e A375444 A(1/6) = 1.35888986768048814311476385141914227984504826245...
%e A375444 where A(1/6)^2 = (3/2)*A(9/64).
%e A375444 A(1/7) = 1.23858760007712401376241920277473621006326963714...
%e A375444 where A(1/7)^2 = (7/5)*A(49/625).
%e A375444 A(1/8) = 1.18621527667665867031082807873688257681814274612...
%e A375444 where A(1/8)^2 = (4/3)*A(4/81).
%e A375444 A(1/9) = 1.15430486498931766438966249826580193821574473318...
%e A375444 where A(1/9)^2 = (9/7)*A(81/2401).
%e A375444 A(1/10) = 1.1323205915354275720071052412999606676975412945...
%e A375444 where A(1/10)^2 = (5/4)*A(25/1024).
%t A375444 terms = 27; A[_] = 1; Do[A[x_]=Sqrt[A[x^2/(1-2*x)^4 ]/(1-2*x)] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Stefano Spezia_, Aug 21 2025 *)
%o A375444 (PARI) {a(n) = my(A=[1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
%o A375444 A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^4 )/(1-2*x) - Ax^2, #A-1) ); A[n+1]}
%o A375444 for(n=0, 30, print1(a(n), ", "))
%Y A375444 Cf. A001006, A000108, A375454, A375443, A375445.
%K A375444 nonn
%O A375444 0,3
%A A375444 _Paul D. Hanna_, Aug 19 2024