A141118 -id:A141118 - cp's OEIS frontend

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

Paul D. Hanna, Jun 05 2008

sign

			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+..

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

Cf. A027436, A141118, A141119, A141121.

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

{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)))}

Define the sequence b(n,m) as follows. If nSeiichi Manyama, May 04 2024

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

Paul D. Hanna, Jun 05 2008

sign

			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 -+ ...

Seiichi Manyama, Table of n, a(n) for n = 1..391
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.

Cf. A027436, A141118, A141120, A141121.

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 *)

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 */

{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)))}

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

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

Paul D. Hanna, Jun 05 2008

sign

			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 +...

Seiichi Manyama, Table of n, a(n) for n = 1..200

Cf. A027436, A141118, A141119, A141120.

{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)))}

From Seiichi Manyama, May 04 2024: (Start)

Define the sequence b(n,m) as follows. If n

A(A(x)) = F(4*x)/4, where F(x) is the g.f. for A141118.

A(A(A(x))) = G(9*x)/9, where G(x) is the g.f. for A027436. (End)

A220288 G.f. A(x) satisfies A(A(A(x))) = x+3x^2+9x^3.

Original entry on oeis.org

1, 1, 1, -8, 28, -26, -386, 2701, -5399, -42155, 358615, -354212, -10419524, 52825312, 236952352, -3103798967, -3013742105, 176201013745, -164790760103, -11763898514324, 27830312919316, 992172068848126, -3681957974446718, -103284064687144985, 528045230825074855

Offset: 1

Dmitry Kruchinin, Dec 09 2012

sign

Seiichi Manyama, Table of n, a(n) for n = 1..200

C. A141118, A220110.

t[n_, m_] := t[n, m] = 1/3*(3^(n - m)* Sum[Binomial[j, n - 3*m + 2*j]*Binomial[m, j], {j, 0, 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}]); t[n_, n_] = 1; Table[t[n, 1], {n, 1, 25}] (* Jean-François Alcover, Feb 22 2013 *)

T(n, m):=if n=m then 1 else 1/3*(3^(n-m)*sum(binomial(j,n-3*m+2*j)*binomial(m,j),j,0,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);

a(n)=T(n,1), T(n, m)=1/3*(3^(n-m)*sum(j=0..m, binomial(j,n-3*m+2*j)*binomial(m,j))-sum(k=m+1..n-1, T(k, m)*sum(, i=k..n, T(n, i)*T(i, k)))-sum(i=m+1..n-1, T(n, i)*T(i, m))), T(n,n)=1.

A372499 G.f. satisfies A(A(A(x))) = F(x), where F(x) is the g.f. for A053540(n) = n*9^(n-1).

Original entry on oeis.org

0, 1, 6, 9, 54, 0, -1944, 44469, -323676, -5990193, 179194032, 484654509, -105337511100, 757846026261, 85419734244300, -1707846638480514, -90276038133498612, 3464956887464464164, 118426852966952180502, -7984363576091338944720, -181143285020960488524558

Offset: 0

Seiichi Manyama, May 03 2024

sign

			A(A(x)) = x + 12*x^2 + 90*x^3 + 594*x^4 + 3807*x^5 + 20412*x^6 + 123201*x^7 + 1032264*x^8 - 1463103*x^9 - 35468766*x^10 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..200

Cf. A309509, A372492.

Cf. A053540, A141118, A372500.

Define the sequence b(n,m) as follows. If n

Showing 1-5 of 5 results.

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

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A141119 G.f. A(x) satisfies A(A(A(A(x)))) = x + 16*x^2.

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A141121 G.f. A(x) satisfies A(A(A(A(A(A(x)))))) = x + 36*x^2.

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A220288 G.f. A(x) satisfies A(A(A(x))) = x+3*x^2+9*x^3.

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A372499 G.f. satisfies A(A(A(x))) = F(x), where F(x) is the g.f. for A053540(n) = n*9^(n-1).

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A220288 G.f. A(x) satisfies A(A(A(x))) = x+3x^2+9x^3.