OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..100
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.
FORMULA
a(n) = T(n,1), where T(n,k) = if n=k then 1 else 1/2*(k*2^(n-k)*binomial(2*n,n-k)/n-sum(i=k+1..n-1, T(n,i)*T(i,k))).
G.f. A(x) satisfies: A( A(x)/(1 + 2*A(x))^2 ) = x. - Paul D. Hanna, Aug 09 2016
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 6*x^3 + 22*x^4 + 90*x^5 + 392*x^6 +...
where A(A(x)) = x*C(2*x)^2 and C(x) is the g.f. of the Catalan numbers;
A(A(x)) = x + 4*x^2 + 20*x^3 + 112*x^4 + 672*x^5 + 4224*x^6 +...
MATHEMATICA
T[n_, n_] = 1;
T[n_, k_] := T[n, k] = 1/2 (k*2^(n-k) Binomial[2n, n-k]/n - Sum[T[n, i] T[i, k], {i, k+1, n-1}]);
a[n_] := T[n, 1];
Array[a, 25] (* Jean-François Alcover, Jul 27 2018, after Vladimir Kruchinin *)
PROG
(PARI) {a(n)=local(A, B, F); F=(1-4*x-sqrt(1-8*x+O(x^(n+3))))/(8*x); A=F; for(i=0, n, B=serreverse(A); A=(A+subst(B, x, F))/2); polcoeff(A, n, x)} /* Paul D. Hanna */
CROSSREFS
KEYWORD
sign
AUTHOR
Vladimir Kruchinin, Nov 11 2011
STATUS
approved