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
A141119
G.f. A(x) satisfies A(A(A(A(x)))) = x + 16*x^2.
Original entry on oeis.org
1, 4, -48, 960, -23296, 616448, -16830464, 456228864, -11849367552, 281940983808, -5672090468352, 75759202861056, 445162740252672, -73915606654517248, 2987936359374651392, -82722417189670879232
Offset: 1
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 -+ ...
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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 *)
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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 */
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{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)))}
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)))}
A372520
G.f. A(x) satisfies A(A(A(A(A(x))))) = Sum_{k>=1} k * 25^(k-1) * x^k.
Original entry on oeis.org
0, 1, 10, -25, 1000, -18125, 131250, 11609375, -630156250, 4314062500, 1173535156250, -38006699218750, -4262573730468750, 321379049072265625, 20787043081054687500, -3209395283374023437500, -116229452332824707031250, 39638105812041778564453125
Offset: 0
Showing 1-4 of 4 results.