A375458 - OEIS

%I #7 Aug 20 2024 09:14:01

%S 1,1,2,6,24,122,756,5532,46696,446748,4778004,56496384,731876160,

%T 10308645192,156862861920,2564468629464,44828643667200,

%U 834403045921936,16476039916372680,344000551441403460,7572120904973371152,175258231266231242316,4255020239957453122152

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

%H Paul D. Hanna, <a href="/A375458/b375458.txt">Table of n, a(n) for n = 1..300</a>

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

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

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

%e G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 24*x^5 + 122*x^6 + 756*x^7 + 5532*x^8 + 46696*x^9 + 446748*x^10 + ...

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

%e RELATED SERIES.

%e A(x+x^2) = x + 2*x^2 + 4*x^3 + 13*x^4 + 54*x^5 + 280*x^6 + 1752*x^7 + 12900*x^8 + 109388*x^9 + 1050222*x^10 + ...

%e A(x-x^2) = x + x^4 + 6*x^5 + 36*x^6 + 240*x^7 + 1836*x^8 + 15996*x^9 + 156726*x^10 + ...

%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 - Ax, #A-1) ); A[n+1]}

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

%Y Cf. A000108.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Aug 19 2024