A030274 - OEIS

%I #27 Jan 13 2024 10:43:00

%S 1,1,1,1,1,0,1,3,-29,25,263,-1481,-5493,80505,41549,-10584341,

%T 14534299,431101045,-1767586509,-43076199745,322525095431,

%U 1295531336537,-30908646610497,-734222129667169,13259294064756895,59705027567272273,-1617292893727823431,-1346735121534484263

%N Numerators of sequence {b(1), b(2), ...} which when COMPOSED with itself gives {1,2,3,...}.

%H Dmitry Kruchinin and Vladimir Kruchinin, <a href="http://arxiv.org/abs/1302.1986">Method for solving an iterative functional equation A^{2^n}(x)=F(x)</a>, arXiv:1302.1986 [math.CO], 2013.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = numerator(T(n,1)), T(n,m) = (1/2)*(binomial(n+m-1,2*m-1) - sum(i=m+1..n-1, T(n,i)*T(i,m))), n > m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 14 2012

%e 1, 1, 1/2, 1/4, 1/8, 0, 1/16, 3/64, -29/128, 25/128, 263/256, -1481/512, -5493/1024, 80505/2048, ... = A030274/A030275

%t t[n_, m_] := t[n, m] = If[ n == m , 1 , 1/2*(Binomial[n+m-1, 2*m-1] - Sum[t[n, i]*t[i, m], {i, m+1, n-1}])]; a[n_] := t[n, 1] // Numerator; Table[a[n], {n, 1, 28}] (* _Jean-François Alcover_, Feb 26 2013, after _Vladimir Kruchinin_ *)

%o (Maxima)

%o T(n, m):=if n=m then 1 else 1/2*(binomial(n+m-1, 2*m-1)-sum(T(n, i)*T(i, m), i, m+1, n-1));

%o makelist(num(T(n, 1)), n, 1, 10); /* _Vladimir Kruchinin_, Mar 14 2012 */

%Y Cf. A030275, A091138.

%K sign,frac

%O 1,8

%A _N. J. A. Sloane_

%E More terms from _Vladeta Jovovic_, Dec 19 2003