A141120 G.f. A(x) satisfies A(A(A(A(A(x))))) = x + 25*x^2.

1, 5, -100, 3250, -127500, 5456250, -241875000, 10733906250, -463469531250, 18897269531250, -699306093750000, 21927485449218750, -487263216796875000, 923644008789062500, 602420821142578125000, -38171197412384033203125

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..128

Formula

Define the sequence b(n,m) as follows. If n<m, b(n,m) = 0, else if n=m, b(n,m) = 1, otherwise b(n,m) = 1/5 * ( 25^(n-m) * binomial(m,n-m) - Sum_{l=m+1..n-1} (b(n,l) + Sum_{k=l..n} (b(n,k) + Sum_{j=k..n} (b(n,j) + Sum_{i=j..n} b(n,i) * b(i,j)) * b(j,k)) * b(k,l)) * b(l,m) ). a(n) = b(n,1). - Seiichi Manyama, May 04 2024

Example

G.f.: A(x) = x + 5*x^2 - 100*x^3 + 3250*x^4 - 127500*x^5 +5456250*x^6+...
A(A(x)) = x + 10*x^2 - 150*x^3 + 4125*x^4 - 140000*x^5 +5162500*x^6+...
A(A(A(x))) = x + 15*x^2 - 150*x^3 + 3375*x^4 - 96250*x^5 +2931250*x^6+...
A(A(A(A(x)))) = x + 20*x^2 - 100*x^3 + 1750*x^4 - 40000*x^5 +918750*x^6+..

Maple

X[1]:= unapply(x+c[2]*x^2, x):
for i from 2 to 6 do
  S:= series((X[i-1]@@5)(x)-x-25*x^2, x, 2^(i-1)+1);
  Sol:=solve({seq(coeff(S, x, k), k=2^(i-2)+1..2^(i-1))}, {seq(c[k], k=2^(i-2)+1
..2^(i-1))});
  X[i]:= unapply(subs(Sol, X[i-1](x))+add(c[j]*x^j, j=2^(i-1)+1..2^(i)), x);
od:
seq(coeff(X[i](x), x, i), i=1..2^5)); # Robert Israel, Jul 20 2020

Prog

(PARI) {a(n, m=5)=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)))}

Crossrefs

Cf. A027436, A141118, A141119, A141121.

Sequence in context: A128784 A369535 A078534 * A123668 A336427 A197200

Adjacent sequences: A141117 A141118 A141119 * A141121 A141122 A141123

Keyword

sign

Author

Paul D. Hanna, Jun 05 2008

Status

approved