A212280 G.f. A(x)=1/(1-F(x)), where F(F(x)) = (1 - sqrt(1-16*x))/8.

1, 1, 3, 17, 131, 1177, 11531, 119201, 1276771, 14015401, 156585211, 1772626673, 20275611347, 233912585849, 2718842818923, 31816917837377, 374657837729987, 4436890509548617

Offset: 0

Author

Vladimir Kruchinin, Feb 14 2013

Keywords

• nonn

Comments

F(x) is the generating function of A213422.

Links

• Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986

Crossrefs

Cf. A213422.

Programs

Maple

T := proc(n,m)
    if n = m then
        1 ;
    else
        m*4^(n-m)*binomial(2*n-m-1,n-1)/n ;
        %-add(procname(n,i)*procname(i,m),i=m+1..n-1) ;
        %/2 ;
    end if;
end proc:
A212280 := proc(n)
    if n = 0 then
        1
    else
        add(T(n,m),m=1..n) ;
    end if;
end proc: # R. J. Mathar, Mar 04 2013

Mathematica

Clear[t]; t[n_, m_] := t[n, m] = 1/2*((m*4^(n-m)*Binomial[2*n-m-1, n-1]/n - Sum[ t[n, i]*t[i, m], {i, m+1, n-1}])); t[n_, n_] = 1; a[n_] := Sum[t[n, m], {m, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Feb 25 2013, from formula *)

Maxima

Solve(k):=block([Tmp,i,j],array(Tmp,k,k),for i:0 thru k do for j:0 thru k do Tmp[i,j]:a,
T(n,m):=if Tmp[n,m]=a then (if n=m then (Tmp[n,n]:1) else (Tmp[n,m]:(1/2*((m*4^(n-m)*binomial(2*n-m-1,n-1))/n-sum(T(n,i)*T(i,m),i,m+1,n-1))))) else Tmp[n,m],  makelist(sum(T(j,i),i,1,j),j,1,k));

Formula

a(n) = sum(m=1..n, T(n,m)) for n>0, where T(n,m)= 1 if n=m, otherwise = (m *4^(n-m) *binomial(2*n-m-1,n-1)/n - sum_{i=m+1..n-1} T(n,i)*T(i,m) )/2.