A199822 G.f. A(x) satisfies A(A(x))=(1-4*x-sqrt(1-8*x))/(8*x).

1, 2, 6, 22, 90, 392, 1772, 8202, 38646, 185076, 900212, 4434356, 22009980, 109780044, 552560376, 2822976810, 14485344790, 72643772868, 361862583908, 2016493563604, 12216275226412, 46909968927072, -57894718593752, 1891831831407844, 70615065586770972

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..100

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

Formula

a(n) = T(n,1), where T(n,k) = if n=k then 1 else 1/2*(k*2^(n-k)*binomial(2*n,n-k)/n-sum(i=k+1..n-1, T(n,i)*T(i,k))).

G.f. A(x) satisfies: A( A(x)/(1 + 2*A(x))^2 ) = x. - Paul D. Hanna, Aug 09 2016

Example

G.f.: A(x) = x + 2*x^2 + 6*x^3 + 22*x^4 + 90*x^5 + 392*x^6 +...
where A(A(x)) = x*C(2*x)^2 and C(x) is the g.f. of the Catalan numbers;
A(A(x)) = x + 4*x^2 + 20*x^3 + 112*x^4 + 672*x^5 + 4224*x^6 +...

Mathematica

T[n_, n_] = 1;
T[n_, k_] := T[n, k] = 1/2 (k*2^(n-k) Binomial[2n, n-k]/n - Sum[T[n, i]  T[i, k], {i, k+1, n-1}]);
a[n_] := T[n, 1];
Array[a, 25] (* Jean-François Alcover, Jul 27 2018, after Vladimir Kruchinin *)

Prog

(PARI) {a(n)=local(A, B, F); F=(1-4*x-sqrt(1-8*x+O(x^(n+3))))/(8*x); A=F; for(i=0, n, B=serreverse(A); A=(A+subst(B, x, F))/2); polcoeff(A, n, x)} /* Paul D. Hanna */

Crossrefs

Cf. A000108.

Sequence in context: A165545 A306023 A150269 * A150270 A049136 A165523

Adjacent sequences: A199819 A199820 A199821 * A199823 A199824 A199825

Keyword

sign

Author

Vladimir Kruchinin, Nov 11 2011

Status

approved