A374568 - OEIS

%I #6 Aug 14 2024 16:37:36

%S 1,2,10,65,508,4534,44836,482214,5571102,68535590,891948664,

%T 12219277950,175517643148,2634970753976,41234024880680,

%U 671089318045929,11337092753702892,198460520744276042,3594448260682257548,67262821868324122100,1298836247390063831188,25850316392778428778716

%N Expansion of g.f. A(x) satisfying x = A(x - x^2) - A(x + x^2)^2.

%C Conjecture: a(n) is odd iff n = 4^k for k >= 0.

%H Paul D. Hanna, <a href="/A374568/b374568.txt">Table of n, a(n) for n = 1..400</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas, in which C(x) = x + C(x)^2 is the Catalan function (A000108).

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

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

%F (3) A(x)^2 = C(-x) + A(-2*C(-x) - x).

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

%F (5) A(x)^2 = A( sqrt(1+4*x) - 1-x ) + (1 - sqrt(1+4*x))/2.

%e G.f. A(x) = x + 2*x^2 + 10*x^3 + 65*x^4 + 508*x^5 + 4534*x^6 + 44836*x^7 + 482214*x^8 + 5571102*x^9 + 68535590*x^10 + ...

%e where x = A(x - x^2) - A(x + x^2)^2.

%e RELATED SERIES.

%e A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 170*x^5 + 1376*x^6 + 12400*x^7 + 122193*x^8 + 1300492*x^9 + 14815264*x^10 + ...

%e where A(x)^2 = C(-x) + A(-2*C(-x) - x) and C(x) is the Catalan function.

%e A(x-x^2) = x + x^2 + 6*x^3 + 37*x^4 + 278*x^5 + 2374*x^6 + 22452*x^7 + 231357*x^8 + 2566806*x^9 + 30395906*x^10 + ...

%e A(x+x^2) = x + 3*x^2 + 14*x^3 + 97*x^4 + 798*x^5 + 7474*x^6 + 77380*x^7 + 869221*x^8 + 10463590*x^9 + 133815278*x^10 + ...

%e where A(x+x^2)^2 = A(x-x^2) - x.

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

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

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

%Y Cf. A000108.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Aug 14 2024