A141118
G.f. A(x) satisfies: A(A(A(x))) = x + 9*x^2.
Original entry on oeis.org
1, 3, -18, 189, -2430, 34020, -486972, 6786261, -86946372, 919825956, -5269375296, -80180038944, 3575424508272, -77211406919844, 1164244485947400, -12342809241883386, 102419678663170128, -2040575112980362980
Offset: 1
G.f.: A(x) = x + 3*x^2 - 18*x^3 + 189*x^4 - 2430*x^5 + 34020*x^6 -+ ...
A(A(x)) = x + 6*x^2 - 18*x^3 + 135*x^4 - 1296*x^5 + 13122*x^6 -+ ...
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T(n,m):=if n=m then 1 else 1/3*(binomial(m,n-m)*9^(n-m)-sum(T(k,m)*sum(T(n,i)*T(i,k),i,k,n),k,m+1,n-1)-sum(T(n,i)*T(i,m),i,m+1,n-1));
makelist((T(n,1)),n,1,7); /* Vladimir Kruchinin, Mar 10 2012 */
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{a(n, m=3)=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)))}
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/* Using Vladimir Kruchinin's formula */
{T(n,k)=if(n==k,1,if(n>k,1/3*(binomial(k,n-k)*9^(n-k) - sum(j=k+1,n-1, T(j,k)*sum(i=j,n, T(n,i)*T(i,j)))-sum(i=k+1,n-1, T(n,i)*T(i,k)))))}
{a(n)=T(n,1)} /* Efficiency can be improved if T(n,k) is stored in an array */
for(n=1,20,print1(a(n),", ")) \\ Paul D. Hanna
A141120
G.f. A(x) satisfies A(A(A(A(A(x))))) = x + 25*x^2.
Original entry on oeis.org
1, 5, -100, 3250, -127500, 5456250, -241875000, 10733906250, -463469531250, 18897269531250, -699306093750000, 21927485449218750, -487263216796875000, 923644008789062500, 602420821142578125000, -38171197412384033203125
Offset: 1
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+..
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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
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{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)))}
A141121
G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = x + 36*x^2.
Original entry on oeis.org
1, 6, -180, 8640, -498960, 31434480, -2055943296, 135216506304, -8720972739072, 538646016002688, -31024094144060160, 1609593032459782656, -71392972690228672512, 2461961564459510280192, -51302015299696881770496, -415041229811424576835584
Offset: 1
G.f.: A(x) = x + 6*x^2 - 180*x^3 + 8640*x^4 - 498960*x^5 +...
A(A(x)) = x + 12*x^2 - 288*x^3 + 12096*x^4 - 622080*x^5 +...
A(A(A(x))) = x + 18*x^2 - 324*x^3 + 11664*x^4 - 524880*x^5 +...
A(A(A(A(x)))) = x + 24*x^2 - 288*x^3 + 8640*x^4 - 331776*x^5 +...
A(A(A(A(A(x))))) = x + 30*x^2 - 180*x^3 + 4320*x^4 - 136080*x^5 +...
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{a(n, m=6)=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)))}
Showing 1-3 of 3 results.