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 A375445 #14 Aug 11 2025 10:20:48
%S A375445 1,1,2,8,41,205,989,4785,23881,124245,673020,3771678,21702164,
%T A375445 127311556,756930002,4539680854,27367146987,165407567379,
%U A375445 1000581963363,6051411131431,36569087782730,220760294880122,1331294835476618,8021165000866546,48296514171243436,290695754850732916
%N A375445 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).
%C A375445 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 A375445 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 A375445 Paul D. Hanna, <a href="/A375445/b375445.txt">Table of n, a(n) for n = 0..400</a>
%F A375445 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A375445 (1) A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).
%F A375445 (2) A(x)^4 = A( x^4*y^5 )*y where y = (1-2*x)^3/((1-2*x)^5 - 2*x^2).
%F A375445 (3) A( x^2*(1 + 2*x)^3 ) = A( x/(1+2*x) )^2 / (1+2*x).
%F A375445 The radius of convergence r satisfies r = (1 - 2*r)^5, where A(r) = 1/(1-2*r) and r = 0.1554302688810578874399658483538386517334...
%e A375445 G.f.: A(x) = 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 A375445 where A(x)^2 = A( x^2/(1-2*x)^5 )/(1-2*x).
%e A375445 RELATED SERIES.
%e A375445 A(x)^2 = 1 + 2*x + 5*x^2 + 20*x^3 + 102*x^4 + 524*x^5 + 2616*x^6 + 13024*x^7 + 66249*x^8 + 348026*x^9 + 1889737*x^10 + ...
%e A375445 A(x)^5 = 1 + 5*x + 20*x^2 + 90*x^3 + 470*x^4 + 2566*x^5 + 13885*x^6 + 74435*x^7 + 400530*x^8 + ... + A375455(n+1)*x^n + ...
%e A375445 SPECIFIC VALUES.
%e A375445 Given the radius of convergence r = 0.15543026888105788743996...,
%e A375445 A(r) = 1.4510850920547193207944317544312912656627353873916...
%e A375445 where r = (1-2*r)^5 and A(r) = 1/(1-2*r).
%e A375445 A(1/7) = 1.273018489928554436323320513425747043274176403249...
%e A375445 where A(1/7)^2 = (7/5)*A(343/3125).
%e A375445 A(1/8) = 1.198855898496093050319216983995020709132914678012...
%e A375445 where A(1/8)^2 = (4/3)*A(16/243).
%e A375445 A(1/9) = 1.160774237134743051625929742274648689798420066384...
%e A375445 where A(1/9)^2 = (9/7)*A(729/16807).
%e A375445 A(1/10) = 1.136139033822992899751347322772302396437733019439...
%e A375445 where A(1/10)^2 = (5/4)*A(125/4096).
%t A375445 terms = 26; A[_] = 1; Do[A[x_] = Sqrt[A[x^2 /(1 - 2x)^5]/(1 - 2x)] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* _Stefano Spezia_, Aug 11 2025 *)
%o A375445 (PARI) {a(n) = my(A=[1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
%o A375445 A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^5 )/(1-2*x) - Ax^2, #A-1) ); A[n+1]}
%o A375445 for(n=0, 30, print1(a(n), ", "))
%Y A375445 Cf. A001006, A000108, A375455, A375443, A375444.
%K A375445 nonn
%O A375445 0,3
%A A375445 _Paul D. Hanna_, Aug 19 2024