A393172 - OEIS

%I #24 Feb 06 2026 09:02:24

%S 1,2,8,42,260,1808,13744,112290,974880,8922276,85580496,856495752,

%T 8913100944,96181220816,1073829057728,12380935349458,147183697957428,

%U 1801623160794480,22680386387491816,293333571234859004,3893918653806360608,53009297608751805168,739458889943201941584,10562209578309277545672

%N G.f. A(x) satisfies A(x-x^2) = x + A(x)^2.

%C Conjecture: for n > 1, a(n) == 2 (mod 4) iff n is a power of 2, otherwise a(n) is divisible by 4.

%H Paul D. Hanna, <a href="/A393172/b393172.txt">Table of n, a(n) for n = 1..520</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A(x-x^2) = x + A(x)^2.

%F (2) A(x*(1-x)*(1-x+x^2)) = x + 2*x*A(x)^2 + A(x)^4.

%F (3) A(x) = C(x) + A(C(x))^2, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

%F (4) A(x) = C(x) + sq(C^2(x) + sq(C^3(x) + sq(C^4(x) + ...))), an infinitely nested square involving successive iterations of the Catalan function C(x) = x + C(x)^2, where sq(x) = x^2 and C^k(x) equals the k-th iteration of C(x).

%F Let A(x)^2 = Sum_{n>=2} b(n)*x^n, then for n >= 2

%F (5.a) b(n) = Sum_{k=1..n-1} a(k) * a(n-k), and

%F (5.b) b(n) = Sum_{k=0..[n/2]} (-1)^k * binomial(n-k,k) * a(n-k), where [n/2] = floor(n/2).

%F (5.c) a(n) = A000108(n-1) + Sum_{k=1..n} k/(2*n-k) * binomial(2*n-k,n-k) * b(k) where A000108(n) = binomial(2*n+1,n)/(2*n+1).

%F (5.d) a(n) = A000108(n-1) + Sum_{k=0..n-1} (n-k)/(n+k) * binomial(n+k,n) * b(n-k).

%e G.f.: A(x) = x + 2*x^2 + 8*x^3 + 42*x^4 + 260*x^5 + 1808*x^6 + 13744*x^7 + 112290*x^8 + 974880*x^9 + 8922276*x^10 + ...

%e RELATED SERIES.

%e The g.f. of A393171 equals A(x-x^2) = x + A(x)^2, where

%e A(x)^2 = x^2 + 4*x^3 + 20*x^4 + 116*x^5 + 752*x^6 + 5328*x^7 + 40644*x^8 + 330324*x^9 + 2838296*x^10 + ... + A393171(n)*x^n + ...

%e A(C(x)) = x + 3*x^2 + 14*x^3 + 81*x^4 + 542*x^5 + 4046*x^6 + 32924*x^7 + 287407*x^8 + 2660942*x^9 + ...

%e A(C(x))^2 = x^2 + 6*x^3 + 37*x^4 + 246*x^5 + 1766*x^6 + 13612*x^7 + 111861*x^8 + 973450*x^9 + ... where A(x) = C(x) + A(C(x))^2.

%e The Catalan function C(x) = x + C(x)^2 begins

%e C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + ... + A000108(n-1)*x^n + ...

%o (PARI) \\ By definition A(x-x^2) = x + A(x)^2

%o {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoef( x + Ser(A)^2 - subst(Ser(A),x,x-x^2), #A-1) ); polcoef(Ser(A),n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A393171 (A(x-x^2)), A000108, A392210.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 04 2026