A141119 - OEIS

%I #27 May 03 2024 11:14:51

%S 1,4,-48,960,-23296,616448,-16830464,456228864,-11849367552,

%T 281940983808,-5672090468352,75759202861056,445162740252672,

%U -73915606654517248,2987936359374651392,-82722417189670879232

%N G.f. A(x) satisfies A(A(A(A(x)))) = x + 16*x^2.

%H Seiichi Manyama, <a href="/A141119/b141119.txt">Table of n, a(n) for n = 1..391</a>

%H Dmitry Kruchinin, 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.

%F a(n) = B(n,1), where B(n,m)=1/2*(T(n,m)-sum(i=m+1..n-1, B(n,i)*B(i,m))), n>m, B(n,n)=1, and where T(n,m)=1/2*(binomial(m,n-m)*16^(n-m)-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. - _Vladimir Kruchinin_, Mar 13 2012

%e G.f.: A(x) = x + 4*x^2 - 48*x^3 + 960*x^4 - 23296*x^5 + 616448*x^6 -+ ...

%e A(A(x)) = x + 8*x^2 - 64*x^3 + 1024*x^4 - 20480*x^5 + 442368*x^6 -+ ...

%e A(A(A(x))) = x + 12*x^2 - 48*x^3 + 576*x^4 - 8960*x^5 + 143360*x^6 -+ ...

%t T[n_, n_] = 1; T[n_, m_] := T[n, m] = 1/2 (Binomial[m, n-m] 16^(n-m) - Sum[T[n, i] T[i, m], {i, m+1, n-1}]);

%t B[n_, n_] = 1; B[n_, m_] := B[n, m] = 1/2 (T[n, m] - Sum[B[n, i]*B[i, m], {i, m+1, n-1}]);

%t Table[B[n, 1], {n, 1, 16}] (* _Jean-François Alcover_, Jul 27 2018, after _Vladimir Kruchinin_ *)

%o (PARI) {a(n, m=4)=local(F=x+m*x^2+x*O(x^n), G); if(n<1, 0, for(k=3, n, G=F+x*O(x^k); for(i=1, m-1, G=subst(F, x, G)); F=F+((-polcoeff(G, k))/m)*x^k); return(polcoeff(F, n, x)))}

%o (Maxima)

%o T(n,m):=if n=m then 1 else 1/2*(binomial(m,n-m)*16^(n-m)-sum(T(n,i)*T(i,m),i,m+1,n-1));

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

%o makelist(B(n,1),n,1,10); /* _Vladimir Kruchinin_, Mar 13 2012 */

%Y Cf. A027436, A141118, A141120, A141121.

%K sign

%O 1,2

%A _Paul D. Hanna_, Jun 05 2008