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A141119
G.f. A(x) satisfies A(A(A(A(x)))) = x + 16*x^2.
4
1, 4, -48, 960, -23296, 616448, -16830464, 456228864, -11849367552, 281940983808, -5672090468352, 75759202861056, 445162740252672, -73915606654517248, 2987936359374651392, -82722417189670879232
OFFSET
1,2
LINKS
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.
FORMULA
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
EXAMPLE
G.f.: A(x) = x + 4*x^2 - 48*x^3 + 960*x^4 - 23296*x^5 + 616448*x^6 -+ ...
A(A(x)) = x + 8*x^2 - 64*x^3 + 1024*x^4 - 20480*x^5 + 442368*x^6 -+ ...
A(A(A(x))) = x + 12*x^2 - 48*x^3 + 576*x^4 - 8960*x^5 + 143360*x^6 -+ ...
MATHEMATICA
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}]);
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}]);
Table[B[n, 1], {n, 1, 16}] (* Jean-François Alcover, Jul 27 2018, after Vladimir Kruchinin *)
PROG
(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)))}
(Maxima)
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));
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));
makelist(B(n, 1), n, 1, 10); /* Vladimir Kruchinin, Mar 13 2012 */
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 05 2008
STATUS
approved