A171199 G.f. satisfies: A(x) = exp( Sum_{n>=1} [A(x)^n + A(x)^-n]*x^n/n ).

1, 2, 3, 8, 25, 83, 289, 1041, 3847, 14504, 55569, 215727, 846761, 3354858, 13398965, 53888063, 218053915, 887107888, 3626373205, 14887942624, 61358959587, 253771944529, 1052917272543, 4381374717994, 18280470530047

Offset

0,2

Comments

Same as A143330 after initial terms.

Links

Table of n, a(n) for n=0..24.

Formula

G.f.: A(x) = (1+x^2 - sqrt(1 - 4*x - 2*x^2 + x^4))/(2*x).

G.f. satisfies: 1 = (A(x) - x)*(1 - x*A(x)).

a(0) = 1, a(1) = 2; a(n) = a(n-1) + a(n-2) + Sum_{k=2..n-1} a(k) * a(n-k-1). - Ilya Gutkovskiy, Jul 20 2021

Example

G.f.: A(x) = 1 + 2*x + 3*x^2 + 8*x^3 + 25*x^4 + 83*x^5 + 289*x^6 +...
log(A(x)) = [A(x)+1/A(x)]*x + [A(x)^2+1/A(x)^2]*x^2/2 + [A(x)^3+1/A(x)^3]*x^3/3 +...

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, (A^m+A^-m+x*O(x^n))*x^m/m))); polcoeff(A, n)}
(PARI) {a(n)=polcoeff((1+x^2-sqrt((1-x^2)^2-4*x+x^2*O(x^n)))/(2*x), n)}

Crossrefs

Cf. A171190, A171191, A143330.

Sequence in context: A127905 A277040 A009224 * A176962 A065619 A243963

Adjacent sequences: A171196 A171197 A171198 * A171200 A171201 A171202

Keyword

nonn

Author

Paul D. Hanna, Dec 05 2009

Status

approved