OFFSET
1,2
FORMULA
G.f. satisfies: A(x) = x*sqrt(1 + 4*A(A(x))).
G.f. satisfies: A(A(x)) = (A(x)^2 - x^2)/(2*x)^2.
a(n)=T(n,1), T(n, m)=m/n*sum(k=1..n-m, sum(i=k..n-m, T(n-m,i)*sum(j=0..i-k, binomial(-k-j+2*i-1,i-1)*(k+j)*2^(k+j)*binomial(k+j-1,k-1))/i*(-1)^(i+k))*binomial(n+k-1, n-1)), n>m, T(n,n)=1. - Vladimir Kruchinin, May 08 2012
EXAMPLE
G.f.: A(x) = x +2*x^2 +6*x^3 + 28*x^4 + 174*x^5 +1308*x^6 +11300*x^7 +...
A(A(x)) = x + 4*x^2 + 20*x^3 +124*x^4 + 912*x^5 +7676*x^6 +72064*x^7 +...
A(x)^2 = x^2 +4*x^3 + 16*x^4 + 80*x^5 + 496*x^6 +3648*x^7 +30704*x^8 +...
PROG
(PARI) {a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x/sqrt(1+4*A +x*O(x^n)))); polcoeff(A, n))}
(Maxima) array(TL, fixnum, 30, 30); T(n, m):=if n=m then 1 else if TL[n, m]=0 then TL[n, m]:m/n*sum(sum(T(n-m, i)*sum(binomial(-k-j+2*i-1, i-1)*(k+j)*2^(k+j)*binomial(k+j-1, k-1), j, 0, i-k)/i*(-1)^(i+k), i, k, n-m)*binomial(n+k-1, n-1), k, 1, n-m) else TL[n, m]; makelist(T(n, 1), n, 1, 15); /* Vladimir Kruchinin, May 08 2012 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 15 2008
STATUS
approved