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 A375443 #11 Jan 16 2025 17:48:09
%S A375443 1,1,2,6,21,77,290,1122,4462,18210,76028,323524,1398071,6115707,
%T A375443 27008516,120162616,537702116,2417043444,10904533054,49343555890,
%U A375443 223851302500,1017798552096,4637127493554,21167261603078,96799606576699,443460169286639,2035144213216892,9355941004378324
%N A375443 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^3 )/(1-2*x).
%C A375443 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 A375443 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 A375443 Paul D. Hanna, <a href="/A375443/b375443.txt">Table of n, a(n) for n = 0..515</a>
%F A375443 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A375443 (1) A(x)^2 = A( x^2/(1-2*x)^3 )/(1-2*x).
%F A375443 (2) A(x)^4 = A( x^4*y^3 )*y where y = (1-2*x)/((1-4*x)*(1-2*x+2*x^2)).
%F A375443 (3) A(x^2 + 2*x^3) = A( x/(1+2*x) )^2 / (1+2*x).
%F A375443 The radius of convergence r satisfies r = (1 - 2*r)^3, where A(r) = 1/(1-2*r) and r = (1/12)*(6 + (6*sqrt(87) - 54)^(1/3) - (6*sqrt(87) + 54)^(1/3)) = 0.20512274384927080786...
%e A375443 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 A375443 where A(x)^2 = A( x^2/(1-2*x)^3 )/(1-2*x).
%e A375443 RELATED SERIES.
%e A375443 A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 58*x^4 + 220*x^5 + 854*x^6 + 3384*x^7 + 13693*x^8 + 56546*x^9 + 237897*x^10 + ...
%e A375443 A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 117*x^4 + 459*x^5 + 1835*x^6 + 7449*x^7 + 30711*x^8 + ... + A375453(n+1)*x^n + ...
%e A375443 SPECIFIC VALUES.
%e A375443 Given the radius of convergence r = 0.2051227438492708078605991264519...,
%e A375443 A(r) = 1.6956207695598620574163671001175353426181793882085...
%e A375443 where r = (1-2*r)^3 and A(r) = 1/(1-2*r).
%e A375443 A(1/5) = 1.51884977058839576453094931523796453209831069839...
%e A375443 where A(1/5)^2 = (5/3)*A(5/27).
%e A375443 A(1/6) = 1.29543251347110009761686143135328534086163706795...
%e A375443 where A(1/6)^2 = (6/4)*A(6/64).
%e A375443 A(1/7) = 1.22025427535592887335278669533719663766721910803...
%e A375443 where A(1/7)^2 = (7/5)*A(7/125).
%e A375443 A(1/10) = 1.12934836581956838019397695630366800332615427708...
%e A375443 where A(1/10)^2 = (10/8)*A(10/512).
%o A375443 (PARI) {a(n) = my(A=[1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
%o A375443 A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^3 )/(1-2*x) - Ax^2, #A-1) ); H=Ax; A[n+1]}
%o A375443 for(n=0, 30, print1(a(n), ", "))
%Y A375443 Cf. A001006, A000108, A375453, A375444, A375445.
%K A375443 nonn
%O A375443 0,3
%A A375443 _Paul D. Hanna_, Aug 18 2024