A213422 G.f. satisfies: A( A(x) - 4*A(x)^2 ) = x.

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

Offset

0,3

Comments

First negative term is a(45).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Dmitry Kruchinin and Vladimir Kruchinin, Method for solving an iterative functional equation A^{2^n}(x)=F(x), arXiv:1302.1986 [math.CO], 2013.

Formula

G.f. satisfies: A(A(x)) = (1 - sqrt(1-16*x))/8.

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

a(n) = T(n,1), T(n,m) = 1/2*(4^(n-m)*m/n*C(2*n-m-1,n-1)-sum(i=m+1..n-1, T(n,i)*T(i,m))), n>m, T(n,n)=1. [Dmitry Kruchinin, Dec 02 2012]

Example

G.f.: A(x) = x + 2*x^2 + 12*x^3 + 96*x^4 + 880*x^5 + 8720*x^6 + 90752*x^7 +...
where
A(A(x)) = x + 4*x^2 + 32*x^3 + 320*x^4 + 3584*x^5 + 43008*x^6 + 540672*x^7 +...+ A000108(n-1)*4^(n-1)*x^n +...
The series reversion of the g.f. A(x) begins:
A(x) - 4*A(x)^2 = x - 2*x^2 - 4*x^3 - 16*x^4 - 80*x^5 - 432*x^6 - 2304*x^7 -...

Mathematica

max = 21; a[0] = 0; a[1] = 1; f[x_] := Sum[a[n]*x^n, {n, 0, max}]; se = Series[f[f[x]] - (1 - Sqrt[1 - 16*x])/8 , {x, 0, max}]; coes = CoefficientList[se, x]; sol = Solve[Thread[coes == 0]]; Table[a[n], {n, 1, max}] /. sol // First (* Jean-François Alcover, Feb 19 2013, from 1st formula *)
T[0, 1]=0; T[n_, n_]=1; T[n_, m_]:= T[n, m]= 1/2*(4^(n-m)* m/n * Binomial[2*n-m-1, n-1] - Sum[T[n, i]*T[i, m], {i, m+1, n-1}]);
a[n_] := T[n, 1];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jan 11 2018, after Dmitry Kruchinin *)

Prog

(PARI) {a(n)=local(A, B, F); A=x+2*x^2; if(n<1, 0, for(i=0, n, B=subst(A, x, A+x*O(x^n)); B=x+4*B^2; F=serreverse(A+x*O(x^n)); A=(A+subst(B, x, F))/2); polcoeff(A, n, x))}
for(n=1, 31, print1(a(n), ", "))
(Maxima)
T(n, m):=if n=m then 1 else 1/2*(4^(n-m)*m/n*binomial(2*n-m-1, n-1) -sum(T(n, i) *T(i, m), i, m+1, n-1));
makelist(T(n, 1), n, 1, 10); [Dmitry Kruchinin, Dec 02 2012]
(SageMath)
@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 A213422(n): return T(n, 1)
[A213422(n) for n in range(31)] # G. C. Greubel, Mar 08 2023

Crossrefs

Cf. A000108, A107700, A307103.

Sequence in context: A371040 A292419 A322543 * A307103 A153231 A365282

Adjacent sequences: A213419 A213420 A213421 * A213423 A213424 A213425

Keyword

sign

Author

Paul D. Hanna, Jun 29 2012

Status

approved