A027436 -id:A027436 - cp's OEIS frontend

A141118 G.f. A(x) satisfies: A(A(A(x))) = x + 9*x^2.

1, 3, -18, 189, -2430, 34020, -486972, 6786261, -86946372, 919825956, -5269375296, -80180038944, 3575424508272, -77211406919844, 1164244485947400, -12342809241883386, 102419678663170128, -2040575112980362980

Offset: 1

Paul D. Hanna, Jun 05 2008

sign

			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 -+ ...

Paul D. Hanna, Table of n, a(n) for n = 1..200

Cf. A027436, A141119, A141120, A141121.

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 */

{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)))}

/* 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

a(n)=T(n,1), T(n,m)=1/3*(binomial(m,n-m)*9^(n-m)-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))), n>m, T(n,n)=1. - Vladimir Kruchinin, Mar 10 2012

A309509 G.f. satisfies A(A(x)) = F(x), where F(x) is the g.f. for A001787(n) = n*2^(n-1).

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 0, 4, 6, -58, 100, 1052, -5924, -21972, 322020, 332392, -21168682, 29068598, 1724404180, -7070346036, -172304798980, 1290100381724, 20728501384592, -247269172883976, -2936888518668676, 53037176259027580, 477640220538178184

Offset: 0

Vladimir Reshetnikov, Aug 05 2019

sign

A(x) is sometimes called a functional square root, or half-iterate of F(x).

Seiichi Manyama, Table of n, a(n) for n = 0..488
Gottfried Helms, Coefficients for fractional iterates exp(x)-1.
Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$, arXiv:1302.1986 [math.CO], 2013.
Wikipedia, Functional square root.

Cf. A001787, A027436, A030274, A030275, A184011.

half[q_] := half[q] = Module[{h}, h[0] = 0; h[1] = 1; h[n_Integer] := h[n] = Module[{c}, c[m_Integer /; m < n] := h[m]; c[n] /. Solve[q[n] == Sum[k! c[k] BellY[n, k, Table[m! c[m], {m, n - k + 1}]], {k, n}]/n!, c[n]][[1]]]; h]; a[n_Integer] := a[n] = half[Function[k, k 2^(k-1)]][n]; Table[a[n], {n, 0, 26}]

Define the sequence b(n,k) as follows. If nSeiichi Manyama, May 03 2024

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

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)

A107099 G.f. satisfies A(A(x)) = x + 4x^3, where A(x) = Sum_{n>=0} a(n)x^(2*n+1).

Original entry on oeis.org

1, 2, -6, 36, -266, 2028, -13596, 50088, 566694, -16598580, 232284876, -1912070088, 631155132, 239439857272, -2781218767224, -17362458802992, 795693633448710, -458070639409908, -335724554310292548, 4520379769156382616, 109439050270732883028, -3828757746830590219608

Offset: 0

Paul D. Hanna, May 13 2005

sign

Coefficients [x^n] A(x) = 0 (mod 3) except at n = 3^k (conjecture).

			A(x) = 1*x + 2*x^3 - 6*x^5 + 36*x^7 - 266*x^9 + 2028*x^11 - 13596*x^13 +-...

Seiichi Manyama, Table of n, a(n) for n = 0..433

Cf. A027436, A097090.

b(n) = local(A,B,F);F=x+4*x^3+x*O(x^n);A=F;if(n==0,0, for(i=0,n,B=serreverse(A);A=(A+subst(B,x,F))/2);polcoeff(A,n,x));
a(n) = b(2*n+1);

A307103 G.f. A(x) satisfies x = A( A(x) + 4*A(x)^2 ).

Original entry on oeis.org

0, 1, -2, 12, -96, 880, -8720, 90752, -975936, 10737152, -120093056, 1360051456, -15556087296, 179424700416, -2084953411584, 24393551634432, -287204585508864, 3400978267127808, -40480500900446208, 484006813958356992, -5810240353159839744, 70001749695581061120

Offset: 0

Michael Somos, Mar 24 2019

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Composition inverse of A027436.

			G.f. = x - 2*x^2 + 12*x^3 - 96*x^4 + 880*x^5 - 8720*x^6 + 90752*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..485

Cf. A027436, A213422.

a[ n_] := Module[ {A, x}, A = x; Do[ A += x O[x]^k; A = Normal[A] + x^k ((-4)^(k-1) CatalanNumber[k-1] - SeriesCoefficient[ ComposeSeries[A, A], k])/2, {k, 2, n}]; Coefficient[A, x, n]];
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0, 0, If[k==n, 1, 2^(2*n - 2*k-1)*(k/n)*Binomial[2*n-k-1, n-1] - (1/2)*Sum[T[n, n-j-1]*T[n-j-1, k], {j,0,n-k-2}] ]]];
a[n_]:= (-1)^(n+1)*T[n,1];
Table[a[n], {n,0,30}] (* G. C. Greubel, Mar 08 2023 *)

{a(n) = my(A); A = x; for(k=2, n, A += x*O(x^k); A = truncate(A) + x^k * ((-4)^(k-1) * binomial(2*k-2,k-1)/k - polcoeff(subst(A, x, A), k))/2); polcoeff(A, n)};

@CachedFunction
def T(n,k):
    if (k<0 or k>n): return 0
    elif (n==0): return 0
    elif (k==n): return 1
    else: return 2^(2*n-2*k-1)*(k/(2*n-k))*binomial(2*n-k, n) - (1/2)*sum( T(n, n-j-1)*T(n-j-1, k) for j in range(n-k-1) )
def A307103(n): return (-1)^(n+1)*T(n,1)
[A307103(n) for n in range(31)] # G. C. Greubel, Mar 08 2023

a(n) = (-1)^(n+1) * A213422(n).

Showing 1-7 of 7 results.

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A141118 G.f. A(x) satisfies: A(A(A(x))) = x + 9*x^2.

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A309509 G.f. satisfies A(A(x)) = F(x), where F(x) is the g.f. for A001787(n) = n*2^(n-1).

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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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A107099 G.f. satisfies A(A(x)) = x + 4*x^3, where A(x) = Sum_{n>=0} a(n)*x^(2*n+1).

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A307103 G.f. A(x) satisfies x = A( A(x) + 4*A(x)^2 ).

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A107099 G.f. satisfies A(A(x)) = x + 4x^3, where A(x) = Sum_{n>=0} a(n)x^(2*n+1).