A213422 -id:A213422 - cp's OEIS frontend

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

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

sign

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).

A212280 G.f. A(x)=1/(1-F(x)), where F(F(x)) = (1 - sqrt(1-16*x))/8.

Original entry on oeis.org

1, 1, 3, 17, 131, 1177, 11531, 119201, 1276771, 14015401, 156585211, 1772626673, 20275611347, 233912585849, 2718842818923, 31816917837377, 374657837729987, 4436890509548617

Offset: 0

Vladimir Kruchinin, Feb 14 2013

nonn

F(x) is the generating function of A213422.

Dmitry Kruchinin, Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986

Cf. A213422.

T := proc(n,m)
    if n = m then
        1 ;
    else
        m*4^(n-m)*binomial(2*n-m-1,n-1)/n ;
        %-add(procname(n,i)*procname(i,m),i=m+1..n-1) ;
        %/2 ;
    end if;
end proc:
A212280 := proc(n)
    if n = 0 then
        1
    else
        add(T(n,m),m=1..n) ;
    end if;
end proc: # R. J. Mathar, Mar 04 2013

Clear[t]; t[n_, m_] := t[n, m] = 1/2*((m*4^(n-m)*Binomial[2*n-m-1, n-1]/n - Sum[ t[n, i]*t[i, m], {i, m+1, n-1}])); t[n_, n_] = 1; a[n_] := Sum[t[n, m], {m, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Feb 25 2013, from formula *)

Solve(k):=block([Tmp,i,j],array(Tmp,k,k),for i:0 thru k do for j:0 thru k do Tmp[i,j]:a,
T(n,m):=if Tmp[n,m]=a then (if n=m then (Tmp[n,n]:1) else (Tmp[n,m]:(1/2*((m*4^(n-m)*binomial(2*n-m-1,n-1))/n-sum(T(n,i)*T(i,m),i,m+1,n-1))))) else Tmp[n,m],  makelist(sum(T(j,i),i,1,j),j,1,k));

a(n) = sum(m=1..n, T(n,m)) for n>0, where T(n,m)= 1 if n=m, otherwise = (m *4^(n-m) *binomial(2*n-m-1,n-1)/n - sum_{i=m+1..n-1} T(n,i)*T(i,m) )/2.

cp's OEIS Frontend

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

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