OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Gi-Sang Cheon, S.-T. Jin, L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.
FORMULA
Given g.f. A(x), then series reversion of B(x)=x*A(x^3) is -B(-x).
Given g.f. A(x), then y=x*A(x^3) satisfies y=x+(xy)^2/(1-(xy)^3).
G.f. satisfies: A(x) = 1 + x*A(x)^2/(1 - x^2*A(x)^3). - Paul D. Hanna, Jun 06 2012
G.f. satisfies: A(x) = 1/A(-x*A(x)^3); note that the Catalan function C(x) = 1 + x*C(x)^2 (A000108) also satisfies this condition. - Paul D. Hanna, Jun 06 2012
a(n) = Sum_{i=0..n/2}((binomial(n+2*i+1,i)*Sum_{k=0..n-2*i}(binomial(k,n-k-2*i)*(-1)^(n-k)*binomial(n+k+2*i,k)))/(n+2*i+1)). - Vladimir Kruchinin, Mar 07 2016
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k-1,k) * binomial(2*n-k+1,n-2*k)/(2*n-k+1). - Seiichi Manyama, Aug 28 2023
PROG
(PARI) {a(n)=local(A); if(n<0, 0, A=x+O(x^4); for(k=1, n, A=x+subst(x^2/(1-x^3), x, x*A)); polcoeff(A, 3*n+1))}
(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+x*A^2/(1-x^2*A^3)); polcoeff(A, n)} \\ Paul D. Hanna, Jun 06 2012
(Maxima)
a(n):=sum((binomial(n+2*i+1, i)*sum(binomial(k, n-k-2*i)*(-1)^(n-k)*binomial(n+k+2*i, k), k, 0, n-2*i))/(n+2*i+1), i, 0, n/2); /* Vladimir Kruchinin, Mar 07 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Sep 20 2005
STATUS
approved